Bertini's Theorem on Generic Smoothness

Home , Hyperplane section, Smoothness, Smooth scheme, Stalk (sheaf)

U.F.R. Mathematiques´ et Informatique Universite´ Bordeaux 1 351, Cours de la Liberation´

Master Thesis in Mathematics

BERTINI1S THEOREM ON GENERIC SMOOTHNESS

Academic year 2011/2012

Supervisor: Candidate: Prof.Qing Liu Andrea Ricolfi ii Introduction

Bertini was an Italian mathematician, who lived and worked in the second half of the nineteenth century. The present dissertation concerns his most celebrated theorem, which appeared for the ﬁrst time in 1882 in the paper [5], and whose proof can also be found in Introduzione alla Geometria Proiettiva degli Iperspazi (E. Bertini, 1907, or 1923 for the latest edition). The present introduction aims to informally introduce Bertini’s Theorem on generic smoothness, with special attention to its recent improvements and its relationships with other kind of results. Just to set the following discussion in an historical perspec- tive, recall that at Bertini’s time the situation was more or less the following:

¥ there were no schemes,

¥ almost all varieties were deﬁned over the complex numbers,

¥ all varieties were embedded in some projective space, that is, they were not intrinsic.

On the contrary, this dissertation will cope with Bertini’s theorem by exploiting the powerful tools of modern algebraic geometry, by working with schemes defined over any field (mostly, but not necessarily, algebraically closed). In addition, our varieties will be thought of as abstract varieties (at least when over a field of characteristic zero). This fact does not mean that we are neglecting Bertini’s original work, containing already all the rele- vant ideas: the proof we shall present in this exposition, over the complex numbers, is quite close to the one he gave. By the way,

iii iv the interested reader can find a detailed description of the original Bertini’s theorems in the beautiful and accurate paper [15] by S. L. Kleiman. Unless otherwise specified we will work with varieties over an arbitrary algebraically closed field k. Bertini’s theorem asserts (in its weak form) that the generic hyperplane section of a smooth projective variety preserves the original smoothness; moreover, the set of hyperplanes satisfying this condition forms itself a quasiprojective variety (a behavior recalling us that parameter spaces of algebraic varieties are usually algebraic varieties themselves). For convenience, we informally say that a (local) property Q of k-varieties is Bertini (or weak-Bertini) if it satisfies condition (B) below and it is strong-Bertini if it satisfies condition (SB):

n (B): If X Pk is a variety satisfying Q then the general hyperplane section of X satisfies Q. (SB): Let X be a variety with a finite dimensional linear system S on it and let X 99K S be the corresponding rational map. If this map induces separable extensions on residue fields (of closed points), then the generic member of S satisfies Q, away from the base points of S and the points in X that are not Q.

Our aim is to show that Q smoothness is (B) and to con- vince ourselves, with some examples rather than by way of precise proofs, that it is also (SB). For most of the time, however, we will be concerned with the weaker (B), so we will become familiar with hyperplane sections in any characteristic; those hyperplanes H giving a smooth intersection with X will be called good, otherwise they will be called bad. We will also show that in characteristic zero smoothness is (SB) (note that the condition of separability is automatically satisﬁed). With an explicit example and a reference to a theorem of Kleiman, we hope to clarify that the inseparability of the extensions between residue ﬁelds is exactly the obstruction for smoothness to become (SB) in positive characteristic. Let us precise that smoothness is not the unique ”Bertini type” property. Suppose, indeed, you have a property Q on your variety X Pn. Then you may ask yourself several questions:

1. Can I ﬁnd at least one hyperplane H such that XXH satisﬁes Q? (sometimes no such one exists) v

2. How many H are there such that X X H satisﬁes Q? (sometimes few such exist)

3. Does the answer to the previous questions depend on the characteristic of the base field? (sometimes positive characteristic makes life hard) Here are examples of Q for which one knows that (B) holds: being reduced, irreducible, normal, Cohen-Macaulay, smooth; and of course it is very interesting to try to enlarge this list as much as possible. It is noteworthy that a property Q has a remarkable hope of being (B) if it is constructible (and the base field is not too small - finite, for example). This will be made clear later but, for the moment, just observe that a constructible set of an irreducible Zariski space either contains an open subset, or is nowhere dense. In what follows we attempt to give a quick description of some more recent results regarding Bertini’s Theorem in its several materializations. Some of them will be discussed later in this work, while others just appear in this introduction as a cue for the reader. The main directions along with the theorem can be improved are:

I. Bertini theorems over Fq; II. study on new Bertini properties;

III. deduce some new results in Algebraic Geometry;

IV. Bertini over a ring of integers.

Let us analyze each of them a little more carefully. § I. The problem with finite fields. With finite fields, everything is finite! The number of rational points in the projective space, n the set of hyperplanes... When k is any field, and X Pk is a smooth quasiprojective variety of dimension m ¥ 0, then one may consider the set t P n¦ | X ¡ p q u U u Pk X Hu is smooth of dimension m 1 over k u n where Hu Pkpuq is the hyperplane corresponding to the point u, defined over the residue field kpuq. It turns out that U is dense n¦ in Pk . If k is infinite, then one can find a hyperplane in U which is defined over k. Otherwise, if k Fq, all the finitely many hyperplanes H over k might give a singular intersection X X H. It is a question by N. M. Katz to see what happens if one considers hypersurfaces of degree greater than one, instead of just looking vi

r s at hyperplanes. Let S be the homogeneous ring Fq x0,..., xn of n P P , and for every homogeneous f Sh d¥0 Sd let Hf be the hypersurface Proj S{f Pn. Deﬁne the density

7 P X Sd µpPq lim dÑ8 7 Sd for any subset P Sh. Then Poonen in [18] (2004), answering to Katz’s question, proves that if X Pn is smooth and quasiprojective of dimension m ¥ 0 over Fq, then the subset

P t f P Sh | X X Hf is smooth of dimension m ¡ 1 u

¡1 has positive density, equal to ζXpm 1q , where ζX is the zeta function attached to the variety X. So, Poonen makes Bertini’s statement become true over Fq by allowing hypersurfaces to play the role of classical hyperplanes. Another interesting result in characteristic p ¡ 0 is due to E. Ballico. In 2003 he proves (see [4]) what follows. He lets k be the algebraic closure of Fp, but in fact he focuses on good n hyperplanes H Pk defined over Fq, where q is a power of p. n In his theorem, X Pk is an irreducible variety of dimension m and degree d. He proves that if q ¥ dpd ¡ 1qm then there exists n 1 some hyperplane H Pk defined over Fq and transversal to X. Note that X might not be defined over Fq. However, the result still holds if X is smooth and defined over Fq: there still exists a good hyperplane cutting X transversally at all of its k-points (and not just at Fq-points): this means that for every x P X one has TX,x H.

§ II. New Bertini properties and generalizations. Let us first focus on the problem of finding new Bertini properties. There exist two properties that we will deal with in this paragraph: weak- normality (WN), and property WN1. Here and in the following a reduced variety X will be said to be WN if every birational universal homeomorphism Y Ñ X is an isomorphism.2 Instead, WN1 means WN with the extra condition that the normalization morphism X˜ Ñ X is unramified in codimension one. One always has that WN1 ñ WN, and they agree in characteristic zero

1 R ¦ n¦ This means that H X Pk . If X is smooth, H is transversal to X if and P only if TX,x H for all x X. If it is not smooth and H cuts it transversally, then P TX,x H for all x Xsm. 2 Equivalently, X coincides with its weak normalization, the maximal couple pXˆ, fˆq among birational universal homeomorphisms Y Ñ X. A universal homeomorphism is a Zariski homeomorphism f such that for every y P Y the extension kpfpyqq kpyq is purely inseparable (so it is trivial when char k 0). vii or for varieties of dimension one. Cumino, Greco and Manaresi showed (see [6]) that Q WN is (B) for varieties of every dimension in characteristic zero. By means of an ”axiomatic approach” to Bertini properties, the same authors show (see [7]) that in fact WN is (SB), again in characteristic zero. The strategy is as follows. They consider local properties Q of noetherian schemes, satisfying three particular axioms (A1), (A2), (A3): for instance, Q reduced, normal, regular all happen to satisfy them. Afterwards, they prove (Theorem 1 in [7]) that if Q satisﬁes these axioms then Q is (SB). One of their results asserts that:

• WN veriﬁes (A1) in characteristic zero;

• WN veriﬁes (A2) and (A3) in any characteristic.

So in characteristic zero, weak-normality is strong-Bertini. In positive characteristic, the authors show that WN does not always satisfy (A1). But this does not imply yet that WN is not strong- Bertini. However, this is the case: they show (see [8]) that, in general, WN is not even (B), weak-Bertini, if char k p. The main theorem in [8] states that:

n Let X Pk be projective, WN, equidimensional, of dimension at least 2. If the general hyperplane section X X H is WN1 then X is WN1 as well.

If X is as above, but is not WN1, then (the authors show that) n intersecting X with a suitable linear subspace L Pk (possibly n X the whole Pk ) gives a weakly normal variety Y X L, and the generic hyperplane section of Y is not WN. The conclusion is that weak-normality is really a property whose ”Bertini-behavior” depends on the characteristic of the base ﬁeld. An important generalization of Bertini’s Theorem is a result nowadays known as Kleiman-Bertini Theorem. It appeared in [16], in 1974. Even if we will discuss it in Chapter 2, it is worth men- tioning its content now. We work over k k of characteristic zero; the situation (not the most general we will deal with) is as follows: one is given of a homogeneous space pX, Gq and of two subvarieties Y, Z X. Then one can consider the translates sY, for s P G. As a variety, it is isomorphic to Y, but it embeds in X differently, namely, the point y goes to its translate sy, viewed inside X. The theorem says that if Y and Z are both smooth then a general translate of, say, Y meets Z transversally: sY XZ is smooth for general s P G. viii

§ III. New ”non-Bertini” results. Bertini’s Theorem has several immediate consequences. The easiest ones will be described in Chapter 2. In [1], the book Arithmetic Geometry, Milne presents a proof of the (highly nontrivial) fact that any abelian variety over an infinite field is a quotient of a jacobian variety. This proof is based on a repeated use of Bertini’s Theorem: one starts with an abelian variety A (of dimension at least 2), and the strategy is to intersect it as many times as dim A¡1 with good hyperplanes. At every step, one finds a smooth variety of dimension one less than at the preceding step, so it is possible to apply Bertini again, by always referring to the same fixed embedding of A into some Pn. Via this procedure, one obtains a curve C A and a morphism from its jacobian J to A; the task reduces finally to showing that such a map is surjective.

§ IV. Bertini in Arithmetic Geometry. The bravest generalization of Bertini’s Theorem is perhaps that shown in a work of P. Autissier (see [2], 2001). In this case it is not even necessary to work over a field and the base scheme is just B Spec OK, where K is a number field. The focus is now the investigation of arithmetic varieties X over OK, and here - as for finite fields - the good locus might not contain any B-rational point. In this case there are no hyperplanes cutting X transversally. Autissier proves (more than) the following: starting with X of dimension d ¥ 3, after 1 a suitable extension L{K one finds an arithmetic variety X over 1 1 ¡ B Spec OL, closed in XOL , such that X has dimension d 1 (it is in fact a hyperplane section of the original X) and for every P 1 p B such that the fiber Xp is smooth, the fiber Xq is also smooth for every q above p. Contents

Introduction iii

1 Linear systems 1 1.1 Coherent and Quasi-Coherent Sheaves ...... 2 1.2 Divisors and Line Bundles ...... 7 1.3 Linear Systems ...... 11 1.4 Comments before the proofs ...... 17 1.5 Bertini in any characteristic ...... 21 1.6 Bertini in characteristic zero ...... 25 1.6.1 Universal Families ...... 27 1.6.2 Constructible Properties ...... 29

2 Applications and Pathologies 39 2.1 First positive results ...... 39 2.2 Something goes wrong: Frobenius ...... 42 2.3 Smooth Complete Intersection ...... 43 2.4 Moving singularities: failure in char 2 ...... 50 2.5 Kleiman-Bertini Theorem ...... 53 2.6 Abelian varieties are quotients of jacobians . . . . . 57 2.7 Arithmetic Bertini Theorem ...... 59

A Appendix 67

Bibliography 79

ix x Contents CHAPTER 1

Linear systems

In this chapter several proofs of Bertini’s Theorem will be provided, in the modern language of linear systems. Two different statements will be proved: the ﬁrst one only holds over the complex numbers, while the second one will turn out to be less general than the previous, but it holds for any characteristic of the (algebraically closed) base ﬁeld k. Here are the statements.

Theorem (Bertini 1) Let X be a smooth complex variety and let D be a positive dimensional linear system on X. Then the general element of D is smooth away from the base locus BD. That is, the set

t H P D | DH is smooth away from BD u

is a Zariski dense open subset of D. n Theorem (Bertini 2) Let X Pk be a smooth projective variety over k. Then the set of hyperplanes n X H Pk such that X H is a smooth scheme is a Zariski n¦ dense open subset of Pk .

Before giving any proof, we want to focus on the notion of linear system as it is understood nowadays, and we wish to explain the precise meaning of the word general. We begin here to apologize for confusing this word with its cousin generic. The subtle difference will be explained later. In this chapter, all that precedes the section named ’Linear Systems’ has to be thought of as background material, which we include by a pure clarity need.

1 2 Linear systems

We start with a brief introduction to quasi-coherent and coherent sheaves on a noetherian scheme X. These objects correspond, locally, to modules and finitely generated modules respectively, under a certain functor. A quasi-coherent sheaf is constructed Coh X by glueing together (sheaves of) modules, exactly as a scheme is ...... made up by glueing together (spectra of) rings...... For the definition of linear system, one is most interested in QCohX ...... line bundles. The smallest abelian category containing line bundles . . . . Coh QCoh ..... (resp. locally free sheaves) is X (resp. X): this is the ...... motivation for the first section. OX-Mod Afterwards, we will relate divisors and line bundles by using sheaf cohomology (that we have available, because our objects are regarded in abelian categories). After the definition of linear system and some remarks, several proofs of Bertini’s Theorem will be provided, and a particular attention will be devoted to the case where the base field is of characteristic zero.

1.1 Coherent and Quasi-Coherent Sheaves

In the ﬁrst part of this section pX, OXq is any scheme and all sheaves are sheaves of modules.

For any OX-module F and for any x P X, one can consider the p q ¢ Ñ p q ÞÑ bilinear morphism β : F X OX,x Fx sending s, tx sxtx. p q b Ñ It induces a natural morphism F X OXpXq OX,x Fx.

Deﬁnition 1.1.1. An OX-module F is said to be generated by its P p q b Ñ global sections at x X if F X OXpXq OX,x Fx is surjective. If this happens for every x P X then one says that F is generated by its global sections. In other words, for every point x P X the module Fx is generated over OX,x by a set of germs of F coming from F pXq.

We will sometimes refer to the following:

Proposition 1.1.1. An OX-module F is generated by its global sections if and only if it is an epimorphic image of a free OX- module, i.e. there exists a set I together with a surjective mor- pIq phism of OX-modules OX F . If F is generated by a subset S F pXq, that is, for every x P X the set t sx | s P S u generates Fx over OX,x, then one can just take I S.

Proof. See [17], Lemma 1.3, p. 158. 1.1. Coherent and Quasi-Coherent Sheaves 3

Deﬁnition 1.1.2. An OX-module F is said to be quasi-coherent if every x P X has an open neighborhood U such that there is an exact sequence of OX-modules pJq| Ñ pIq| Ñ | Ñ OX U − OX U − F U − 0.

Quasi-coherence is a property of local nature on X. Informally, one could translate the deﬁnition as follows: a sheaf is quasi- coherent if, locally, it is the cokernel of an OX-morphism between free sheaves.

Deﬁnition 1.1.3. An OX-module F is said to be finitely generated if every point x P X has an open neighborhood U such that, for some n ¥ 1, there is a surjective morphism of OX-modules n| | OX U F U.

Definition 1.1.4. An OX-module F is said to be coherent if it n| Ñ | is finitely generated and every OX-morphism OX U F U has finitely generated kernel for any open subset U. The property of being generated by global sections is independent of coherence. For instance, M is globally generated but it might not be coherent. And any nonzero invertible sheaf of negative degree is coherent but is not generated by its global sections. In particular, a finitely generated sheaf need not be globally generated. A quasi-coherent sheaf is completely determined by the local Some important equivalences affine data, that is, a sheaf of modules F on a scheme X is quasi- of categories. coherent if and only if on every open affine subset U Spec A, there is an isomorphism F |U F pUq , where is the functor sending an A-module M to the sheaf M associated to M on Spec A. This is a fully faithful exact1 functor, whose essential image is QCohU. Moreover, F is coherent whenever the same holds true, and in addition F pUq is finitely presented over A (or just finitely generated, when A is noetherian). In fact, when re- stricted to finitely presented A-modules, the equivalence A-Mod QCohU restricts to an equivalence with the category CohU. It should be clear from these characterizations that any locally free OX-module is quasi-coherent, and every locally free OX-module of finite rank is coherent.2 However, there are examples showing that the converse of these assertions is false: for instance, the

1 Not only it is exact, but it also reﬂects exactness of sequences of the shape M Ñ L Ñ N. 2 These assertions are not true for any scheme, but certainly hold for those X such that the structure sheaf OX is coherent; in particular, X locally noetherian is enough. 4 Linear systems

sheaf M on X Spec A, where M is not a free A-module, is quasi- coherent but not locally free.

Remark 1.1. While proving the above characterizations of coherence, one realizes that for a quasi-coherent OX-module F the following implications are true: Coherent ñ Finitely generated ñ F pUq finitely generated for every U open affine. Moreover, these conditions are equivalent whenever X is noetherian. Remark 1.2. The OX-module M, the quasi-coherent sheaf on X Spec A associated to M, is an example of a sheaf that is generated by its global sections: indeed any set I of generators of M will pIq provide a surjective morphism of OX-modules OX M. An- other way to see this: just apply the definition and observe that for every p P X the canonical map M bA Ap Ñ Mp defined by m b pa{sq ÞÑ pmaq{s is surjective, because m{s comes from the tensor m b p1{sq.

We now introduce a characterization of quasi-coherence. First, by the local nature of this notion, for knowing a quasi-coherent sheaf on X it is a good start to know its sections over the affine open subsets U Spec A X. In fact, even better is to know how they restrict from U to any of its distinguished affine open subsets, i.e. those of the form Spec Af for f P A. We call the open immersion Spec Af ãÑ Spec A a distinguished inclusion. So, consider an OX-module F and define

F pSpec Aqf F pSpec Aq bA Af. Now consider the following diagram

res ... F pSpec Aq ...... F pSpec Afq ...... ¡ b ...... A ...... A f ...... φ ......

F pSpec Aqf

Clearly by ¡ bA Af we mean the image of ` : A Ñ Af under F pSpec Aq bA ¡. Note that F pSpec Aq is an A-module and F pSpec Afq is an Af-module, so by the universal property of localization there exists a unique morphism φ closing the diagram: for every distinguished inclusion we have a canonical factoriza- tion of the restriction map. The following statement holds true. 1.1. Coherent and Quasi-Coherent Sheaves 5

Proposition 1.1.2. An OX-module F is quasi-coherent if and only if for every such diagram the canonical morphism φ is an isomorphism. The following result will be useful.

Theorem 1.1. Let X be a scheme.

p1q The category of quasi-coherent OX-modules is abelian.

p2q If F and G are quasi-coherent OX-modules, then so too is b F OX G , and we can compute its sections on every open affine subset U X simply by forgetting sheafification, that p b qp q p q b p q is F OX G U F U OXpUq G U .

Proof. Part p1q: the claim is certainly true in the affine case. For any scheme X, one knows that QCohX OX-Mod is a subcategory, with OX-Mod abelian. So one only needs to check that QCohX contains the 0 object, is closed under finite direct sums, kernels and cokernels. The 0 sheaf is certainly quasi-coherent. If F and G are two quasi-coherent OX-modules, consider an affine open r subset Spec A of X and assume that F M and G N on Spec A. Then F ` G pM ` Nq because, by exactness of localization, one has pM ` Nq pDpfqq pM ` Nqf Mf ` Nf for every f P A. So F ` G is again quasi-coherent. Let us consider a morphism α : F Ñ G of quasi-coherent sheaves. One has to be sure it has both a kernel and a cokernel. Let U be any affine subset of X. Use quasi-coherence: on U, the morphism α is given as an OXpUq- linear map of modules β : M Ñ N. Define pker αqpUq ker β and pcoker αqpUq coker β. Now, if β ...... 0 ...... LMNP...... 0 is an exact sequence, then so is the localized sequence β ...... f ...... 0 ...... Lf ...... Mf ...... Nf ...... Pf ...... 0 for every f P OXpUq. So pker βqf kerpβfq and pcoker βqf cokerpβfq. Hence, thanks to Proposition 1.1.2, both pker βqf and pcoker βqf define quasi-coherent OX-modules, and moreover these are exactly the kernel and the cokernel of α, as it is clear by checking on stalks. Part p2q: the question is local on X so one may assume X U Spec A; suppose that F pUq M and r G pUq N, two A-modules (so that F M and G N). Then p b qp q p b r qp q p b qp q b F OX G U M OX N U M A N U M A N, as claimed. 6 Linear systems

The end of this section is devoted to the deﬁnition of the quasi- coherent sheaf associated to the Proj of a graded ring and to twisted sheaves, which will be used from now on. Now S is a graded ring and M a graded S-module. Denote X Proj S. One can prove that there is a unique sheaf on X, denoted M, such that

| p q M D pfq Mpfq (1.1) for every homogeneous f P S . Equivalently, we define it on the principal basis of X by MpD pfqq Mpfq, the submodule of Mf consisting of elements of degree 0. This M is a quasi-coherent OX- module, by (1.1) and by our characterization in Proposition 1.1.2. If S is noetherian and M is finitely generated then it is coherent. For every d P Z one can consider the graded ring Spdq: it is just S (so it is an S-module) with grading Spdqn Sd n. For X Proj S we define OXpdq Spdq . In particular, when d 1 one gets the Twisting Sheaf of Serre. If f P S is homogeneous of d degree one, then Spdqpfq f Spfq, i.e.

0 d 0 H pD pfq, OXpdqq f H pD pfq, OXq. (1.2)

Clearly the OX-modules OXpdq are quasi-coherent, so one can ap- p q b p q p q ply Theorem 1.1 to see that OX d OX OX e OX d e for every two integers d, e. Indeed, for every f P S homogeneous of degree one,

p p q b p qqp p qq OX d OX OX e D f p qp p qq b p qp p qq OX d D f OXpD pfqq OX e D f d p p qq b e p p qq d ep b q f OX D f Spfq f OX D f f Spfq Spfq Spfq d e f Spfq Spd eqpfq OXpd eqpD pfqq.

In fact, the following general result is true:

Theorem 1.2. Let S be a graded ring and X Proj S. If S is generated by S1 as an S0-algebra and if S0 is noetherian, then:

(a) Every OXpdq is an invertible sheaf on X, hence coherent.

(b) For d, e any two integers, OXpdq b OXpeq OXpd eq.

Proof. For (a), one has to show that OXpdq is locally free of rank P one. Take f S1, then by the above p q| p p q q OX d D pfq S d pfq , 1.2. Divisors and Line Bundles 7

the quasi-coherent sheaf on Spec Spfq associated to the S-module p q p q| S d pfq. The claim is that OX d D pfq is free of rank one. This t p q | P u will be enough because D f f S1 is a covering of X by our assumption. Let us show that Spdqpfq is free of rank one over Spfq. In fact, Spfq is the algebra of elements of degree 0 in Sf, and Spdqpfq is the algebra of elements of degree d in Sf (recall that deg f 1). So there is a group isomorphism Spfq Ñ Spdqpfq d deﬁned by s ÞÑ f s. This is well deﬁned for every d P Z since f is invertible in Sf. Hence Spdqpfq is free of rank one over Spfq. Part (b) follows from the calculation above.

Remark 1.3. If X Proj S where S is generated by S1 as an p p qq S0-algebra, then S1 Γ X, OX 1 are global sections generating p q p q S1 p q OX 1 , i.e. they give a surjective OX-morphism OX OX 1 , so for such an S we have another example of a sheaf that is generated by its global sections.

1.2 Divisors and Line Bundles

This section is devoted to introduce the notion of line bundle by highlighting its (cohomological) relations with divisors. This can be done for any variety X (thought of as a space with a sheaf of functions O), but for simplicity we may assume that X is a noetherian integral and normal scheme (so that we have Weil divisors available). By a divisor, one means in fact Cartier divisor, ¢ ¢ i.e. a global section of the quotient sheaf M {O , where M is the sheaf of total quotient rings on X (according to complex man- ifolds, one can call its sections ”meromorphic functions”). Recall that one has the following result:

Theorem 1.3. Let X be an integral, separated and noetherian scheme such that Op is a UFD for every p P X. Then the group ¢ ¢ Div X of Weil divisors is isomorphic to the group H0pX, M {O q of Cartier divisors, and principal Weil divisors correspond to principal Cartier divisors under this isomorphism (so Cl X Ca Cl X).

Proof. See [14] II, Proposition 6.11, p. 141.

Thus, as our interest will be in smooth varieties, we will be almost always allowed to interchange Weil with Cartier.

A divisor D can be identiﬁed with some local data t Uα, fα u, that is, an open covering U pUαq of X together with a collection 8 Linear systems

¢ of nonzero meromorphic functions fα P M pUαq satisfying

fα ¢ P O pUαβq. (1.3) fβ

Here Uαβ denotes the intersection Uα X Uβ. A Cartier divisor is said to be effective in case one can choose the fα to be regular functions. n Recall, as an example, that divisors on Pk are completely determined, up to linear equivalence, by their degree, and the de- n gree map induces an isomorphism Cl Pk Z. We will see that linear equivalence of divisors corresponds to isomorphism between invertible sheaves (line bundles), and this for any scheme X. By a (complex) vector bundle of rank r over X one means a surjective morphism of varieties π : E Ñ X such that for all x P X ¡1 the ﬁber Ex π pxq is a C-vector space of dimension r, and there exists an open neighborhood U of x together with an isomorphism (called trivialization)

r φ : E|U −Ñ U ¢ C

¡1 where E|U π pUq and Ey E|U is sent isomorphically to r t y u ¢ C for every point y P U.

Theorem 1.4. Let X be a variety and r a positive integer. Vec- tor bundles of rank r over X correspond, up to isomorphism, to locally free OX-modules of rank r.

Proof. See Appendix.

Giving a vector bundle pE, πq is, more or less,3 the same as giving a collection of transition functions t gαβ : Uαβ Ñ GLpr, Cq u, ¢ i.e. a one-cochain in the Cechˇ complex C pU, O q. Explicitly, each gαβ is a morphism of varieties satisfying gαβpxqgβαpxq 1 for all x P Uαβ and the so-called cocycle condition

gαγpxq gαβpxqgβγpxq @ x P Uαβγ. (1.4)

A line bundle is just a vector bundle of rank 1. Let us give ourselves a complex line bundle L Ñ X, so that its ¢ transition functions take values in GLp1, Cq C . Let U pUαq ¢ be an open covering of X and let fα P O pUαq be nonzero regular | Ñ ¢ functions. If φα : L Uα Uα C are trivializations then the

3 See (:) to justify this more or less. 1.2. Divisors and Line Bundles 9

¢ corresponding transition functions gαβ : Uαβ Ñ C are given by p q p ¥ ¡1q| gαβ x φα φβ Lx where ¡1 φβ| φ | Lx .. α Lx .. t u ¢ ...... t u ¢ x C . Lx . x C.

More precisely, to get gαβpxq one has to evaluate this composi- tions at px, 1q and look at the second component. Now consider the new trivializations ψα : fαφα (these are again isomorphisms by the choice of fα). The corresponding transition functions hαβ are deﬁned by h pxq pψ ¥ ψ¡1q| ψ | ¥ φ |¡1 αβ α β Lx α Lx β Lx f pxq p q | ¥ p ¡1p q |¡1q α p ¥ ¡1q| fα x φα Lx fβ x φβ L φα φβ Lx x fβpxq whence the relation fα hαβ gαβ. (1.5) fβ Of course, we have changed trivializations (and hence transition functions), but L is still the same. On the other hand, there is no other way to ”deform” a given collection of trivializations without getting another vector bundle (here another means not isomorphic), therefore our conclusion is:

Two one-cocycles t gαβ u and t hαβ u determine the same line bundle (up to isomorphism) if and only if ¢ there are nonzero holomorphic functions fα P O pUαq (:) t ¡1 u satisfying (1.5), if and only if their difference gαβhαβ is a Cechˇ one-coboundary. This leads us to the relation 1 ¢ LX H pX, O q : Pic X (1.6) where LX denotes the set of line bundles over X, up to isomorphism. This is really a group isomorphism. By Theorem 1.4, we can view Pic X as the group of isomorphism classes of invertible sheaves on X.

Deﬁnition 1.2.1. To a Cartier divisor D t Uα, fα u on a scheme X one can associate the following subsheaf L pDq of M : for every α, we set L pDqpUαq to be the OpUαq-submodule p qp q ¡1 p q p q L D Uα fα O Uα M Uα . ¢ Since fα{fβ P O pUαβq, we have that fαOpUαβq fβOpUαβq, so that L pDq is a well deﬁned object; it is called the invertible sheaf associated to D. As a notation, we may also write OpDq instead of L pDq. 10 Linear systems

Remark 1.4. Note that each divisor on X can be viewed as a closed subscheme D X, and the inclusion is given by the sheaf of ideals OXp¡Dq. In fact, there is an exact sequence

0 −Ñ OXp¡Dq −Ñ OX −Ñ OD −Ñ 0.

Aside 1.1. Note that, when X is a regular curve over k with function ﬁeld K, the global sections of this invertible sheaf are

0 H pX, OXpDqq LpDq

where LpDq denotes the k-vector space t f P K¢ | pfq D ¥ 0 u Y t 0 u. In this case, the meaning of the space LpDq is clear: it contains those p q p q ¥ ¡ rational functions φ whose order° ordp φ vp φ np at every p q point of X, if D is the divisor p npp: the functions in L D are those whose poles have order no worse that np. For an arbitrary scheme X over a ﬁeld k, even if Weil divisors do not coincide with Cartier divisors, we 0 always have this k-vector space H pX, OXpDqq. Now it contains those ¢ meromorphic functions φ P ΓpX, M q such that the divisor pφq D is an effective Cartier divisor, i.e. such that pφq ¥ ¡D. This means: take pφq to be the image of φ under p¦ (see below); sum (multiply) by D 0p ¢{ ¢q in H X, M OX ; look at the local data corresponding to this product and check whether they are nonzero regular functions. Note, ﬁnally, that when Weil Cartier, the concept of effective Weil divisor is the same as that of effective Cartier divisor: indeed, a rational function has positive valuation along a prime divisor Z if and only if it is regular (i.e. it has no poles) on Z.

Proposition 1.2.1. Let pX, Oq be a scheme and D, E be Cartier divisors on X. Then ¢ ¢ H0pX, M {O q (a) The morphism L induces a bijection ...... ¢ ¢ . 0 . H pX { q t u . , M O Invertible Subsheaves of M up to ...... ¡ Pic X (b) We have L pD ¡ Eq L pDq b L pEq 1, and

Proof. The first point is clear: you cannot have an invertible sub- ¢ sheaf H M without a collection fα P M pUαq: simply look at ¡1 local generators hα of H , and take their inverses fα hα to be the local data of some divisor D. To prove (b) just observe that, ¢ ¢ H0pX, M {O q being a group, D ¡ E is defined by the quotient of ¡ the defining functions, and thus it goes exactly to L pDqbL pEq 1. For the last part, L pDq P Pic X is trivial if and only the transition functions t hαβ u of the corresponding line bundle are in 1 ¢ B pU, O q, if and only if hαβ are quotients of holomorphic functions, i.e. meromorphic functions defined on the open subsets 1.3. Linear Systems 11

Uα. And the datum of these meromorphic functions is equivalent to the datum of a single global nonzero meromorphic function ¢ f P M pXq, which means exactly that D is the divisor of f. So D is principal if and only if it is in ker L . Conclude by (b).

Remark that the short exact sequence

¢ ¢ p ¢ ¢ 1 −Ñ O −Ñ M −Ñ M {O −Ñ 1 gives an exact piece in cohomology

p¦ 0 ¢ ...... 0 ¢ ¢ ...... L ...... H pX, M q .... H pX, M {O q .... Pic X and moreover one has an injective group homomorphism

...... θ : Ca Cl X ...... Pic X. which becomes an isomorphism when, for example, X is an integral scheme (See [14] II, Proposition 6.15, p. 145 for a proof). As our varieties will be integral, we will assume that a line bundle will be of the form L pDq.

1.3 Linear Systems

From now on, unless otherwise stated, k denotes an algebraically closed field, and X is an integral smooth projective variety over k. Let us briefly summarize what we know: first of all, for such an X, Weil Divisors coincide with Cartier divisors (consequence of Theorem 1.3); second, linear equivalence of divisors corresponds to isomorphism of invertible sheaves (consequence of Proposition 1.2.1); moreover, as X is integral, there is an isomorphism Pic X Ca Cl X, saying that a line bundle L on X can be written as OXpDq for some divisor D on X. Here is another important and nontrivial result that one has to keep in mind.

Theorem 1.5. Let X be a projective scheme over a noetherian 0 ring A and let F be a coherent OX-module. Then H pX, F q is a ﬁnitely generated A-module.

Proof. [14] II, Theorem 5.19, p. 122.

The underlying idea of a linear system is the following: giv- 0 ing an invertible OX-module L and a set S H pX, L q of global sections is the same as giving a certain collection of effective divisors, all linearly equivalent to each other. Let us make this precise: to an invertible sheaf L on X and a nonzero global section 12 Linear systems s P H0pX, L q, one associates its divisor (called the divisor of zeros), denoted by psq (or sometimes psq0, to emphasize it has no negative part). It is deﬁned as follows: for every point η P X such ¡ that Yη t η u is irreducible of codimension one, one has the cyclic module Lη e.OX,η and the germ sη writes in the form P p q p q e.u for a unique u OX,η. One deﬁnes vη s vη u , where the vη in the right hand side is the discrete° valuation associated to the p q p q prime divisor Yη. Finally, s : η vη s Yη is clearly effective. Proposition 1.3.1. Let X be a smooth projective variety over k and let D be a divisor on X. Consider OXpDq L pDq, the corresponding invertible sheaf. Then

0 (a) For every nonzero global section s P H pX, OXpDqq, the divisor psq0 is an effective divisor linearly equivalent to D. (b) Every effective divisor linearly equivalent to D is the divisor 0 of zeros of some s P H pX, OXpDqq.

1 0 (c) Two sections s, s P H pX, OXpDqq share the same divisor of 1 ¢ zeros if and only if s λs for some λ P k .

Proof. For (a), use that OXpDq is a subsheaf of the sheaf M of meromorphic functions on X, that now is the constant sheaf K. So ¢ s can be viewed as a rational function f P K . Assume t Ui, fi u ¢ are local data defining D, where fi P K . Then, by the construc- ¡1 tion of OXpDq, we know it is locally generated by f . Notice i that there is a local isomorphism φ : OXpDq −Ñ OX, indeed p qp q ¡1 p q p q OX D Ui fi OX Ui is isomorphic to OX Ui via multiplica- ¡1 ÞÑ p q t u tion by fi, i.e. fi h h. Thus s 0 is locally defined by Ui, fif , i.e. psq0 D pfq, showing that psq0 D. For (b), let E ¥ 0 be an effective divisor such that E D pfq. Then 0 pfq ¥ ¡D, which is equivalent (by Aside 1.1) to f P H pX, OXpDqq. Finally, the divisor of zeros of f is exactly pfq0 E. 1 1 For (c), suppose psq0 ps q0 where, as above, s and s correspond 1 ¢ 1 1 to some f, f P K , and we have pf{f q 0, as f{f is a meromor- { 1 P p q 0p ¢q phic function. This says that f f ker Γ X, p¦ H X, OX , but now X is a projective variety over an algebraically closed field, 0 1 ¢ hence H pX, OXq k, and then f{f λ P k .

If |D| is the set of all effective divisors linearly equivalent to D, the Proposition above says that |D| has the structure of a projective space over k. More precisely, there is a bijective map 0 ¢ ...... pH pX, OXpDqq ¡ t 0 uq{k .... |D|

...... rss. ... psq D. 1.3. Linear Systems 13

Recall: the projectivization of a vector space V, denoted PV, is by definition the projective variety Proj pSym Vq; one can think of it as the set of hyperplanes in V; it is in fact the dual construction ¢ of pV ¡t 0 uq{k , the set of lines through the origin in V. It follows ¢ ¦ by the definition that one always has pV ¡ t 0 uq{k PV . Thus one may assume the following identification:

0 ¢ 0 ¦ pH pX, OXpDqq ¡ t 0 uq{k PH pX, OXpDqq .

0 ¦ Thus, the points of the projective variety PH pX, OXpDqq act as parameters for the set of divisors in |D|.A point of |D| is a one- 0 dimensional subspace of H pX, OXpDqq, namely, such a line consists of those global sections which differ one from each other by a nonzero constant, that is, having the same divisor of zeros. Deﬁnition 1.3.1. The set |D|, together with its structure of projective variety, is called a complete linear system of divisors on X. A linear system on X is a subset D |D| which is a linear subspace of |D| when this is viewed as a projective variety. In other ¦ 0 words, D PV for some vector subspace V H pX, OXpDqq. ¦ The dimension of a linear system PV is dimk V ¡ 1, so it is ﬁ- nite by Theorem 1.5. A linear system of dimension one is called a pencil, while in dimension two it is called a net and in dimension three a web. Remark 1.5. Why does one projectivize the dual? Because to any 0 inclusion V H pX, OXpDqq there corresponds a surjective mor- 0p p qq¦ ¦ 4 phism H X, OX D V and since Proj is contravariant one ¦ recovers an inclusion PV ãÑ |D|. In what follows, however, one ¦ will identify V to its dual, so PV will mean the linear system PV .

One can figure a linear system as a continuous family of vari- n eties, moving in some bigger ambient variety, say Pk ; the points that do not move at all when one ranges the set of parameters are the points in the base locus, which we now define properly. Definition 1.3.2. A point x P X is called a base point of D if it lies in Supp D for every D P D, where the support Supp D X of a divisor D is the union of its prime divisors. We define £ BD Supp D DPD to be the base locus of D. In other words, a point x P X is a base point if it lies in every divisor contained in the system. A divisor

4 Proj is actually not a functor, but it behaves like a functor when one restricts to arrows (of graded rings) that are epimorphisms. 14 Linear systems

E which is contained in the base locus, i.e. such that E ¤ D for all D P D, is called a fixed component of D. n ¡ Example 1.3.1. Let X Pk . For every integer d 0 the invertible sheaf Opdq has the space of homogeneous polynomials of degree ¨ n d d as global sections: it is a k-vector space of dimension d . n One can consider, for an arbitrary hyperplane H Pk , the¨ com- | | 0p n p qq n d ¡ plete linear system dH PH Pk , O d of dimension d 1. It is the projective variety of effective divisors linearly equivalent to dH (that is, hypersurfaces of degree d), and it gives an example of a base-point-free linear system. As a notation, one might also denote |dH| by |Opdq|.

Example 1.3.2. Let X be a smooth projective curve over k with function ﬁeld K, and let D P Div X. Consider the complete linear system |D| associated to the k-vector space

¢ LpDq t φ P K | pφq D ¥ 0 u Y t 0 u .

¢ ¢ That is, |D| PLpDq t φ P K | pφq D ¥ 0 u {k is a projective space of dimension `pDq ¡ 1. A point of this projective space is a ¢ class rφs t aφ | a P k u where all the aφ share the same pφq0. The correspondence is then

|D| PLpDq pφq D ÐÑ rφs ° p q p q r s where φ p ordp φ p. Saying that φ is in the base locus amounts to asserting that pφq D ¤ Dλ for all Dλ P |D|, i.e. all points (prime divisors) p appearing in pφq D also appear in every Dλ. Note that if D is (the divisor of) a point p then |p| t p u, 1 unless X Pk, in which case every point is linearly equivalent to | | 1 P 1 each other, so that p P for every point p Pk.

Lemma 1.1. Let D PV |D| be a linear system on X. Then x P X is a base point of D if and only if sx P mxOXpDqx for all s P V. In particular, D is base-point-free if and only if OXpDq is generated by the global sections in V. t P | P p q¢ u Proof. If Xs y X sy OX D y , we can write £ £ £ BD Supp D Supp psq pX ¡ Xsq, DPD rssPD sPV since the support of psq psq0 is exactly the set of points at which s vanishes. It follows that a point x P X is a base point if sx R p q¢ P P p q P OX D x for any s V. That is, sx mxOX D x for all s V. 1.3. Linear Systems 15

Suppose the above condition is false for any x P X, i.e. D is base- point-free. This is equivalent to saying that for any x P X there is a global section s P V such that sx does not vanish (is invertible) in OXpDqx. Such sections are the required generators of OXpDqx over OX,x.

The sheaf OXpDq is not generated by its global sections in general, so a complete linear system might have base points. For instance, consider X a smooth curve of genus ¡ 0 and D the divisor of any point. As remarked above, |p| t p u. It follows from the lemma (and from Proposition 1.1.1) that one can also deﬁne the base locus in this way: t P | b Ñ p q u BD x X V k OX,x OX D x is not surjective .

We need two quick observations. ¡ n The dual projective space. We know that Pk is the set of vector lines of kn 1. If one applies the same construction to ¦ the dual vector space pkn 1q then the result is the space of all n-dimensional linear subspaces (hyperplanes) of kn 1. This is n¦ called the dual projective space and is denoted Pk . Of course, every n-dimensional linear subspace of kn 1 is of the form ¤ ¤ ¤ Ha : a0x0 a1x1 anxn 0 P p q where not all the ai k are zero, so that a a0, a1,..., an n is a well deﬁned point of Pk . Moreover Ha Hb if and only if P ¢ n Hb λHa for some λ k , if and only if a b in Pk . This remark n¦ n ÞÑ allows us to identify Pk to Pk through Ha a. Summarizing,

n¦ t n u 0p n p qq Pk Hyperplanes in Pk PH Pk , O 1 .

Finally, note that the vacuous statement ’the points on a hyper- n plane H Pk form a hyperplane’ gets dualized to ’the hyper- P n n¦ planes through a point p Pk form a hyperplane in Pk ’. ¡ The ”generic element”. Let D Pr be a linear system of divisors on X.A general member of D is said to satisfy a property Q if there is a Zariski dense open subset U Pr such that all divisors corresponding to points of U satisfy Q. The generic element of a linear system is the generic point of the projective space Pr parameterizing the system, and a given property is called generic if it is a property of the generic point. Tricky observation: recall that the generic point of an irreducible scheme is unique, so the contrary of the phrase ’the generic point 16 Linear systems

ξ satisﬁes Q’ is simply ’ξ does not satisfy Q’. But if one is not necessarily dealing with schemes, and has the assertion

’The general member of D satisﬁes Q’, (1.7) then this has a more subtle negation. Call VQ D the set of divisors satisfying Q. The contrary of (1.7) is then:

’VQ has empty interior’.

r This translates the fact that VQ does not contain any open set of P , that is to say: there is no dense open subset of Pr parameterizing divisors satisfying Q. Finally, note the relation between generic and general: a ﬁrst remark is that it is not possible to deduce some generic information from a general one, in the sense that if some property Q is true for general points, we cannot deduce that Q is generic; in the reverse direction, however, one can go

generic ; general in this way: let Q be a given property and let X Ñ Y be a morphism, where Y is an irreducible scheme with generic point ξ. One then checks whether the generic ﬁber Xξ has Q; afterwards, if Q is constructible, then there is an open neighborhood U of ξ where the property holds, that is, Xy has Q for all y P U. Let us point out that sometimes one confuses generic with general. Just to make an example: when one says ’a normal variety is gener- ically regular’, it means that there is a nonempty open subset where all the points are regular. Now we are set to begin our discussion on Bertini’s Theorem. We give two statements.

Theorem (Bertini 1) Let X be a smooth complex variety and let D be a positive dimensional linear system on X. Then the general element of D is smooth away from the base locus BD. That is, the set

t H P D | DH is smooth away from BD u (1.8)

1.4 Comments before the proofs

As a statement, the first one is more general (the associated line bundle might not be very ample, in particular not globally generated), but only holds in characteristic zero. The second one, instead, describes the complete base-point-free linear system associated to Op1q. Let us see in detail why (Bertini 2) is a particular case of (Bertini 1), of course when char k 0 is assumed. The first step is showing that, for any linear system D PV Pr |D|, of dimension r, one has a morphism ψ ... X ¡ BD ...... D. 0 Fix a basis t e0,..., er u V H pX, OXpDqq of V. For any P ¡ b Ñ p q x X BD the map σx : V k OX,x OX D x is surjective, that is, t | P u p q sx s V generates OX D x e.OX,x over OX,x. One can write p q P p q P the germs ei x e.ui OX D x for some (unique) ui OX,x. At least one ui is invertible in the ring OX,x, because if every ui P were in the maximal ideal mx OX,x then, for every s V, one would have sx P mxOXpDqx, that is, σx would not be surjective, contradicting our assumption. So, define

ψpxq pe0pxq,..., erpxqq pu0pxq,..., urpxqq.

This is well defined because if we choose another local generator 1 p q 1 P e of OX D x, say e e.v for some (unique) v OX,x, then p 1 p q 1 p qq p p q p q p q p qq p p q p qq e0 x ,..., er x e0 x v x ,..., er x v x e0 x ,..., er x . This was the first step. Note that the definition of ψ depends on the choice of a basis for V. Now start with the closed immersion ãÑ n n¦ 0p n p qq i : X Pk and recall that Pk PH Pk , O 1 is a complete n linear system on Pk of dimension n. Write r s{ r s X Proj k x0, x1,..., xn a Proj k y0, y1,..., yn (1.9) where yi xi mod a. If IX is the sheaf of ideals corresponding ãÑ n to i : X Pk , there is an exact sequence

−Ñ −Ñ n −Ñ i¦ −Ñ 0 IX OPk OX 0.

By a twist, that is, applying ¡ b Op1q, one ﬁnds another exact5 sequence

0 −Ñ IXp1q −Ñ Op1q −Ñ OXp1q −Ñ 0

5 Tensoring by an invertible sheaf is an exact functor, because the question is local and, locally, this sheaf is OX. 18 Linear systems

where by OXp1q we mean pi¦OXqp1q. Notice that, since i is an immersion, we have pi¦OXq|X OX, so pi¦OXqp1q|X OXp1q. By n taking global sections on Pk and then restricting to global sections on X one gets 0p n p qq Ñ 0p n p qp qq Ñ 0p p qq H Pk , O 1 − H Pk , i¦OX 1 − H X, OX 1 . 0 Let α denote this composition and let V im α H pX, OXp1qq. One can show - this is the second step - that the linear system D PV |OXp1q| is base-point-free, i.e. for all x P X the set t | P u p q sx s V generates OX D x over OX,x. Once this is done, one can deduce by (Bertini 1) that the general member of D is smooth (away from the empty set!), and the statement about (1.8) becomes exactly (Bertini 2). 0p n p qq Let us verify that PV is base-point-free. As H Pk , O 1 is generated by the n 1 global sections x0,..., xn, the map α is deﬁned by xi ÞÑ yi (but notice that, as global sections, the yi’s might be linearly dependent over k). If we let Ui D pyiq then we see that Ui X X D pxiq form an open (afﬁne) covering of X. Moreover

p q| p q x n | p q p q| y | O 1 D xi iOPk D xi implies that OX 1 Ui iOX Ui . P p q Thus for every x Ui one has OX 1 x yiOX°,x; in fact, to avoid p q the dependance on i one may write OX 1 x i yiOX,x, but this now holds for every x P X since pUiq is a covering. This says that b Ñ p q b ÞÑ the canonical map V OX,x OX 1 x sending v u u.vx is surjective for every x P X. So D is base-point-free. In addition, there is a morphism ψ : X Ñ D and one veriﬁes that by composing

... ψ ...... n X .... D .... Pk ãÑ n one recovers the closed immersion i : X Pk one started with. Again, it is enough to check this fact locally on the covering pUiq. In fact, to the sequence of ring homomorphisms ...... p q ...... r s ...... r s O Ui k y0,..., yn pyiq k x0,..., xn pxiq there corresponds the sequence of morphisms of afﬁne schemes

. ...... r s ...... r s Ui Spec k y0,..., yn pyiq Spec k x0,..., xn pxiq which is exactly the local translation of what we want. In particular, in the above situation the morphism ψ is a closed immersion, and dim X ¤ dim D.

The existence of the morphism ψ : X ¡ BD Ñ D allows to formulate another deﬁnition of the base locus, that we put in the shape of a proposition, as follows. 1.4. Comments before the proofs 19

Proposition 1.4.1. The base locus of a linear system D on X is the subset of X at which the rational map X 99K D is not deﬁned. Proof. Assume D PV |D|. The above map is not deﬁned at x P X if and only if all the sections (in a basis) of V vanish on x. This happens if and only if they all lie in mxOXpDqx, which means that the sections in V do not generate OXpDq. Conclude by Lemma 1.1.

Example 1.4.1. If X is a smooth curve, then any linear system on X is base-point-free, because - more generally - for a curve C, any n rational map C 99K V, for a subvariety V Pk , is deﬁned at all the smooth points.

Remark 1.6. Denote again by ψ the morphism X Ñ Pr associated to some base-point-free D. Then, if one wants to see the effective divisors in D as subvarieties of X, it is enough to consider the ﬁbers of ψ. It means that ¡ ¦ D t ψ 1pHq | H P Pr u .

This will become clear after Lemma 1.2. But if one accepts the fact ¦ that Σ t px, Hq P X ¢ Pr | ψpxq P H u, together with the projec- ¦ tion π : Σ Ñ Pr , is a family (parameterizing the divisors in D) ¡ with members the ﬁbers π 1pHq, then we can already understand ¢ ¡1p q ¡1p q this remark because X Pr H ψ H π H . Finally, what ¦ Bertini says is just that there is a dense open subset U Pr such ¡1 that ψ pHq ¡ BD is smooth for every H P U.

Example 1.4.2. Let k be an algebraically closed ﬁeld. Consider 1 the 3-uple Veronese embedding of Pk, that is

...... 1 ...... 3 Pk .. Pk . .. p q ...... 3 2 2 3 t, u . . pt , t u, tu , u q 3 3 Its image C Pk is called the twisted cubic curve in Pk. We claim 3 that a smooth projective curve X Pk of degree 3 such that X 1 is not contained in a hyperplane and X Pk as abstract algebraic varieties is of the form X σ C for some automorphism P p q 3 σ PGL 3, k of Pk. By viewing such an X as a closed subscheme 3 of Pk, we know that we can recover this closed immersion by

...... i ...... 3 X ...... P .. k ...... D 20 Linear systems

0p 3 p qq 0p p qq where D PV and V im H Pk, O 1 H X, OX 1 . By our assumption that X is not contained in any hyperplane, we 0p 3 p qq have that α is injective, thus V H Pk, O 1 , a k-vector space of dimension 4. So dim D 3. In addition, D has degree6 deg D 3, 1 and X Pk says that V must correspond to a 4-dimensional vector 0p 1 p qq space W H Pk, O 3 , which is already of dimension 4, so 0p 1 p qq V H Pk, O 3 and D is complete. Since the closed immersion ãÑ 3 i : X Pk is determined by V and the choice of a basis for V (Theorem 1.6 below), we have that i is the 3-uple embedding up 3 to the choice of that basis, i.e. up to a linear automorphism of Pk.

Aside 1.2. We want to say more about the rational map ψ : X 99K D. n | p q| When X Pk , we have¨ the base-point-free linear system O d on X, of n d ¡ dimension m d 1. So we have a morphism

...... n ...... m Pk ... Pk

0p n p qq which is constructed in this way: take any basis of H Pk , O d , for n example the set of monomials of total degree d in the coordinates on Pk . P n m Then x Pk is sent to the point of P whose coordinates are evaluations of the monomials at x, for instance in lexicographic order. This is exactly the d-uple Veronese embedding. The Example 1.4.2 above works with n 1 and d 3; when n d 2, we have the following situation: the 2-uple embedding ...... 2 ...... 5 Pk .. Pk is exactly the morphism corresponding to the base-point-free linear sys- | p q| 2 tem O 2 on Pk, i.e. the complete linear system of conics. Its image is called the Veronese surface.

Before passing to the proofs of Bertini’s Theorem, let us describe the relation between linear systems on a projective variety X and morphisms to projective space. We have the following result.

Theorem 1.6. Let X be a scheme over a ring A. Then p q Ñ n ¦p p qq i If φ : X PA is an A-morphism then φ O 1 is an invertible sheaf on X generated by the n 1 global section ¦ si φ pxiq.

piiq If L is an invertible sheaf on X generated by n 1 global 0 sections s0,..., sn P H pX, L q then there exists a unique Ñ n ¦p p qq A-morphism φ : X PA such that L φ O 1 and

6 The degree of a linear system is the degree of any of its divisors: it is well deﬁned because the degree of a divisor on a smooth projective curve depends only on its linear equivalence class. 1.5. Bertini in any characteristic 21

¦ si φ pxiq for each i 0, . . . , n under this isomorphism of sheaves.

Proof. See [14], II, Theorem 7.1, p. 150.

Ñ n By the above, giving X Pk is the same as giving a set

0 S t s0,..., sn P H pX, L q gererating L , with L invertible u .

Ñ n Now, by Proposition 1.3.1, giving a morphism X Pk is the same as giving a base-point-free linear system PV on X and a basis t s0,..., sn u of V. If another basis is chosen, the new morphism n will differ from the ﬁrst one by an automorphism of Pk . Ñ n If one requires, in addition, that a morphism X Pk be a closed immersion, then one has to add two conditions: the k- vector space generated by s0,..., sn must separate points and tangent vectors. However, if one has recourse to ample line bundles, a characterization of immersions inside the projective space comes almost for free. A line bundle L on a Y-scheme X is said to be ãÑ n very ample if there exists an immersion i : X PY such that ¦ L i Op1q. When Y Spec k, a line bundle L is very ample on a k-scheme X if and only if there are ﬁnitely many global sections 0 s0,..., sn P H pX, L q, generating L , such that the induced mor- n phism to Pk is an immersion. Summary: we have the following equivalences of data.

Maps Linear Systems

n P 0p q X 99K Pk L ; s0,..., sn H X, L .

Ñ n P 0p q X Pk L ; s0,..., sn H X, L generate L .

ãÑ n P 0p q X Pk L very ample; s0,..., sn H X, L generate L .

A divisor D on X is said to be very ample in case OXpDq is very ample. Hence, very ample divisors are exactly those which allow embeddings in the projective space (provided that they are globally generated).

1.5 Bertini in any characteristic

In this section we present the proof of Bertini’s Theorem (Bertini 2), over an arbitrary algebraically closed ﬁeld k. We need to keep in mind the following two results. 22 Linear systems

n ¢ m Ñ m Proposition 1.5.1. The second projection Pk Pk Pk is a closed map.

n Proof. The projective space Pk is proper over a point (Spec k) and properness is closed under base extension. It follows that the second projection is proper, and hence closed.

Proposition 1.5.2. Let X and Y be projective varieties of dimension n and m respectively. Let f : X Ñ Y be a surjective morphism. Then m ¤ n, and

¡ p♥q For any y P Y, every irreducible component of f 1pyq has dimension ¥ n ¡ m.

¡ p♦q There is a nonempty open set U Y such that dim f 1pyq n ¡ m for all y P U.

Proof. See [Shafarevich, I, Theorem 7 on p. 76].

n Theorem 1.7 (Bertini 2). Let X Pk be a smooth projective n variety over k. Then the set of hyperplanes H Pk such that X n¦ X H is a smooth scheme is a Zariski dense open subset of Pk .

Proof. Let d be the dimension of X. We look for a dense open n¦ subset U Pk such that the corresponding hyperplanes H have the good property of letting X X H be a smooth scheme. We may assume that d n ¡ 1. Indeed, if X is a smooth hypersurface, the n¦ ¡ ¦ required good locus is easily seen to be the open subset Pk X X (because x is smooth in X H if and only if H TX,x). Moreover, we may assume that X is not contained in any hyperplane. We deﬁne the set of ”bad hyperplanes” at a closed7 point p P X to be

t P n¦ | X u n¦ Bp H Pk p is singular in X H Pk .

Now, if we let p range the closed points of X, we can deﬁne a set

t p q | P P u ¢ n¦ B p, H p X is a closed point and H Bp X Pk .

Our strategy is the following:

p1q Showing that B has the structure of a projective variety, and

p q Ñ 2 showing that the ﬁrst projection π1 : B X is surjective, and that in this case we have dim B ¤ n ¡ 1.

7 Nonsingularity can be checked on closed points. 1.5. Bertini in any characteristic 23

Suppose these statements are both proved. Let us explain why we can conclude. As dim B n we get that the second projection Ñ n¦ π2 : B Pk sends B onto a proper closed subset of the dual n¦ ¡ p q projective space. We can then deﬁne U Pk π2 B and see that it is the required dense (in particular, nonempty) open subset in the statement. To prove p1q, we note that we have a family (see Deﬁnition ¢ n¦ n¦ 1.6.1) B X Pk of subvarieties of Pk with base X. Concretely, ¤ B t p u ¢ Bp. pPX

¡1 The members of this family are the fibers q ppq Bp where q : B Ñ X is the pullback projection. Hence B B X H, where H is the universal hyperplane (see (1.15)). To prove p2q, it is enough to show that, for every closed point P ¡1p q ¡ ¡ p X, the fiber π1 p is a projective space of dimension n d 1. ¡ ¡ If this is done then π1 is automatically surjective because n d 1 is a strictly positive integer, as we assumed d n¡1. In this case, we can apply Proposition 1.5.2 (♥) to see that for every closed P ¡ ¡ ¡1p q ¥ ¡ point p X we have n d 1 dim π1 p dim B d, hence dim B ¤ n ¡ 1.8 This would conclude the second step, so our last ¡1p q n¡d¡1 P claim is that π1 p P for every closed point p X. To prove our claim, let us fix a closed point p P X, and ho- n n¦ mogeneous coordinates x0,..., xn on Pk and a0,..., an on Pk ; 0p n p qq name W H Pk , O 1 , the k-vector space of homogeneous linear polynomials. Now, there exists a hyperplane that does not pass through p; without loss of generality we may assume it is H0 V px0q. Define a morphism of k-vector spaces

φp ...... { 2 W . OX,p mp ¸n ¸n . xi ...... a x . .... a i i . i x i0 i0 0

This is well deﬁned because x0ppq 0. Note that φp is surjective: indeed, since p is a closed point and k is algebraically closed, the maximal ideal mp OX,p is generated by linear polynomials in { { the coordinates x1 x0,..., xn x0. This says that the image of φp

8 In fact, dim B n ¡ 1. This is not important. However, it follows by Proposition 1.5.2 (♦): there is a nonempty open subset of X such that all its ¡ ¡ ¡1p q ¡ ¡ points y satisfy n d 1 dim π1 y dim B dim X dim B d, which implies dim B n ¡ 1. 24 Linear systems

{ 2 contains a basis of OX,p mp, so φp is surjective. { 2 As X is smooth, dimk mp mp dim X d. Note that

2 OX p{m , p k {m2 d { 2 implies that dimk OX,p p 1. mp mp

Combined with the fact that dimk W n 1 and with surjectivity of φp, it implies

¡ { 2 dimk ker φp dimk W dimk OX,p mp (1.10) pn 1q ¡ pd 1q n ¡ d.

Notice that if f P W and Hf V pfq then, whenever f P ker φp, we have that f vanishes in a neighborhood of p in X, and hence on the whole X (by irreducibility); thus X Hf. Conversely, if X Hf for some f P W then f|X 0 and f P ker φp. Summarizing,

f P ker φp ðñ X Hf. (1.11)

However, this will never happen, by our assumption on X. Next, observe that for f P W our closed point p is in X X Hf if and only if fppq 0, if and only if f is not invertible in a neighborhood of p q P { 2 p, if and only if φp f mp mp. Assume this is the situation and p q 2 P write then φp f yp mp for yp y mp. We can deﬁne the P X { 2 local ring A at p X Hf: if R OX,p mp and a yOX,p, { p q { A OXXHf,p R φp f R OX,p a.

By deﬁnition A is a local ring. When is it regular? The answer will tell us when p is a regular point of XXHf. The maximal ideal of A is mA mp{a and

2 p { q2 p 2 q{ p 2 q{ 2 {p 2 X q mA mp a mp a a mp a mp a mp a .

For the regularity of A, we are interested in comparing the dimen- { 2 sion dim A to dimk mA mA. First, we obtain

2 2 m mp{m mp{m m {m2 p p p A A 2 p 2 q{ 2 {p 2 X q mp a mp a mp a mp a and by computing dimensions we get

{ 2 ¡ {p 2 X q dimk mA mA d dimk a mp a . { By regularity of X at p and by the fact that A OX,p a (where a is principal) we have dim A d ¡ 1. So A is regular if and only 1.6. Bertini in characteristic zero 25

{ 2 {p 2 X q if dim A dimk mA mA, if and only if dimk a mp a 1. This 2 happens exactly when y is not contained in mp:

X ðñ P ¡ 2 p is regular in X Hf yp mp mp (1.12)

Now we can use (1.11) and (1.12) to characterize the bad hyperplanes at p.

Bp t Hf | p is singular in X X Hf, with f P W u t | P 2 P u Hf yp mp, with f W .

P 2 p q P But again, yp mp if and only if φp f 0, i.e. f ker φp. In ¢ addition, it is clear that Hf Hλf for every λ P k , so we can identify Bp Ppker φpq, proving that dim Bp n ¡ d ¡ 1 thanks ¡1p q to (1.10). It remains to observe that π1 p Bp to conclude that ¡1p q ¡ ¡ π1 p is a projective space of dimension n d 1, as claimed. This completes the proof.

1.6 Bertini in characteristic zero

In this section we present a proof of Bertini’s Theorem for a pencil over C, and a proof over a ﬁeld of characteristic zero which is valid for any linear system. The ﬁrst one is quite easy; the second one relies on some fundamental theorems, like the generic smoothness theorem.

Theorem 1.8 (Bertini 1). Let X be a smooth complex variety and let ` be a pencil on X. Then the general element of ` is smooth away from the base locus B. That is, the set

t λ P ` | Dλ is smooth away from B u (1.13) is a Zariski dense open subset of `.

Proof. To any λ P ` P1 there corresponds an effective divisor Dλ on X; let us assume that dim X d and let us work with our pencil locally inside a polydisc ∆ X (the analytic analog of an afﬁne open subset in the Zariski topology). If f, g are linearly independent sections generating ` inside ∆, then every divisor Dλ has the following local representation, in the analytic topology: p q p q Dλ : f z1,..., zd λg z1,..., zd 0 (1.14) where zi are the local coordinates on ∆. Here we are using that a smooth complex algebraic variety has the structure of a complex 26 Linear systems manifold, so we can work locally in the analytic topology. Let us 1 see what happens if for some λ P P (different from 0, 8), pλ is a singular point for Dλ outside the base locus of the system. For all i 1, . . . , d we then have

fppλq λgppλq 0 Bf Bg ppλq λ ppλq 0. Bzi Bzi p q p q P Note that if f pλ 0 g pλ then pλ νPP1 Dν B; but pλ R B so we get that f and g do not both vanish at pλ, hence neither one does, whence for all i 1, . . . , d we must have

fppλq Bf fppλq Bg λ ¡ and ppλq ¡ ppλq 0. gppλq Bzi gppλq Bzi

Now, let us calculate the derivatives of the ratio f{g, and evaluate them at pλ:

¡ © Bf Bg B f B ppλqgppλq ¡ fppλq B ppλq pp q zi zi λ 2 0. Bzi g gppλq

Next, call V the locus of singular points (in ∆) of the divisors in `, and call S the subvariety of ∆ ¢ P1 cut out by the equations

f λg 0 Bf Bg λ 0 Bzi Bzi where λ ranges P1 and i 1, . . . , d. Then V is the image of S inside ∆ (under the pullback projection), that is V is locally cut out by the above equations. But now, by a calculation we made, the ratio f{g is locally constant on V ¡B. This means that here it takes only finitely many values. Translation: there are only finitely many 1 λ P P such that Dλ is singular away from BD (in fact, B X ∆). 1 Thus Dλ is smooth away from the base locus for almost all λ P P , proving that there is a dense open subset U P1 such that the corresponding divisors are smooth away from B X ∆. Of course, we worked locally on ∆, but there are finitely many such polydiscs covering our variety X, so the result follows.

Remark 1.7. Note how by tacit agreement we used the map ψ : X ¡ B Ñ P1. If q is a point outside the base locus, we deﬁne ψpqq to be the unique λ such that q lies on Dλ. This is unique just by the linearity condition of (1.14). In fact the key point in our proof is that ψ|V¡B has ﬁnite image. 1.6. Bertini in characteristic zero 27

Remark 1.8. Perhaps, it is possible to generalize this proof to a linear system of arbitrary dimension. Bertini himself asserts (see [5]) that it is so. But the details are not so easy to handle, and we do not dispose of a precise proof for the moment.

The next subsection is a digression, devoted to formalize what we already said in the Introduction about varieties ”parameterizing” collections of varieties: a motivation for this digression is that a central object in the study of Bertini’s Theorem is the set of n hyperplane sections ΣX of a variety X Pk ; such a set is in fact a variety and, even more, it constitutes an example of a universal n¦ family over Pk . We already encountered this family during the proof of Bertini in any characteristic, and we already used all of its geometric structure. More precisely, we still have to justify the ¢ n¦ fact that the bad hyperplanes form a subvariety B X Pk . We n ¢ n¦ just asserted that B is the intersection (inside Pk Pk ) between Ñ n ¢ n¦ a family B X and the universal hyperplane H Pk Pk . Hopefully, this will become clearer soon.

1.6.1 Universal Families In the following two subsections k is a field of any characteristic, at least until the new proof of Bertini’s Theorem in characteristic zero. It might not be algebraically closed. Some elementary examples of universal families are provided; the most important one, that of parameter space, is omitted, but the reader can find a nice discussion on it in [12]. n We now think of Pk as our big ambient variety. Let B be any p q n variety, and Vb bPB a collection of algebraic varieties Vb Pk . First, we want to answer this question: what does it mean that such a collection varies algebraically and continuously with parameters? Let us give an intuition on that: it is a possible point of view to regard a morphism of schemes π : X Ñ Y as a collection of schemes pXyq, where Xy is the fiber X ¢Y Spec kpyq; such a collection of fibers is thought of as a family of schemes varying algebraically with parameters (the points y P Y) because each fiber is a pullback of two arrows depending only on y. Thus the answer to our question is (partially) inside the following definition.

Deﬁnition 1.6.1. Let B be a variety. A family of projective vari- n ¢ n eties in Pk with base B is a closed subvariety V B Pk together ¡1 with the induced projection π : V Ñ B. The ﬁbers Vb π pbq are called the members of the family.

But what about continuity? The fibers of π might be very 28 Linear systems different one from each other (some might be empty, there can be dimension jumps...). The collection of fibers V pXyqyPY is a family of subschemes of X with base Y. It turns out that the right notion to express the continuity of V is flatness. We see its importance in the definition of a universal family: a family V Ñ 1 1 B is universal if for every flat family V Ñ B whose members 1 belong to V there exists a unique regular map B Ñ B such that 1 1 V B ¢B V . So in particular any universal family is flat. 1. The Universal Hyperplane. Consider the projective variety n¦ n Pk of hyperplanes in Pk , and the subset t p q | P P n¦ u n¦ ¢ n H H, p p H and H Pk Pk Pk . (1.15) n If z denotes the homogeneous coordinates on Pk and w are those n¦ on Pk , then it is clear that H is given by the single polynomial ¸n fpz, wq ziwi. i0 n¦ ¢ n Thus it defines a hypersurface in Pk Pk . If one puts B n¦ Ñ Pk and considers the first projection π1 : H B, then each ¡1p q n hyperplane H has fiber π H H Pk , so H may be viewed n n¦ as a family of hyperplanes in Pk with base Pk . Symmetrically, if n Ñ n one takes Pk as base, and considers the projection π2 : H Pk , P n ¡1p q t p q | Q u then the fiber at a point p Pk is π2 p H, p H p , n¦ which is a hyperplane in Pk ; so in this case H may be viewed n¦ n as a family of hyperplanes in Pk with base Pk . It is called the universal hyperplane because it has the following property:

1 1 ¢ n For every ﬂat family V B Pk of hyperplanes 1 Ñ n¦ there exists a unique regular map B Pk such that 1 1 V B ¢ n¦ H. That is, there is a pullback diagram Pk

1 ... V ...... H ...... ¦ 1 ...... n B .... Pk where the bottom arrow is the unique morphism send- P 1 1 1 ing b B to the member Vb of V . n 2. The Universal Hyperplane Section. Let X Pk be a projec- n tive variety. View H as a family of hyperplanes with base Pk , i.e. Ñ n together with the second projection π2 : H Pk . Deﬁne t p q | P X P n¦ u n¦ ¢ ΣX H, p p X H and H Pk Pk X. 1.6. Bertini in characteristic zero 29

¡1p q n¦ ¢ Then ΣX π2 X is a subvariety of Pk X and hence it is a n¦ family (of subvarieties of X) over Pk . It is called the universal hyperplane section. Note that

X p n¦ ¢ q ΣX H Pk X ,

n¦ ¢ n the intersection being inside Pk Pk . And, most important, the P n¦ X ﬁber at each point H Pk is isomorphic to X H. With the same spirit, one can deﬁne the universal conic C 5 ¢ 2 Pk Pk and the universal line.

1.6.2 Constructible Properties

The next aim is to brieﬂy state some basic facts about constructible subsets of Zariski spaces. A Zariski space is a noetherian topological space such that every closed nonempty irreducible subset has a unique generic point. Afterwards, we introduce constructible properties of morphisms of schemes; the reference for this subject is [EGA, tome IV3]. So in this section all schemes considered are noetherian. To simplify, one may also assume they are algebraic varieties.

Definition 1.6.2. Let X be a Zariski space. A subset E X is said to be constructible if it is a finite disjoint union of locally closed subsets of X. It is called locally constructible if X has a covering X X i Xi such that E Xi is constructible in Xi for every i. If X is any scheme, this is the same as asserting that EXV is constructible for every open affine subset V X.

Remark 1.9. If X is a quasi-compact and quasi-separated scheme (e.g. an algebraic variety), then locally constructible implies constructible.

Proposition 1.6.1. The collection of constructible subsets of X coincides with the collection C characterized by the following properties:

p1q any open subset of X is in C ;

p2q the class C is closed under complements (thus it contains the closed subsets);

p3q the class C is closed under ﬁnite unions (and thus under ﬁnite intersections). 30 Linear systems

Proof. The collection C contains every constructible set, by how it is defined. Conversely, open and closed subsets are constructible, and so are also finite unions of constructible sets, thus constructible subsets satisfy p1q and p3q. So it remains to show that in C we c find² all complements E where E is constructible. In fact, as p p X qqc p X qc 1¤i¤r Ui Fi 1¤i¤r Ui Fi and a finite intersection of locally closed is locally closed, it is enough to prove that Ec is constructible for E locally closed. Let us write E U²X F, where U is open and F is closed. Then Ec Fc Y Uc Fc pUc ¡ Fcq, which is a union of an open set with a closed subset, hence both constructible.

Deﬁnition 1.6.3. Let X be a topological space. A subset E X is said to be nowhere dense (in french: rare) if X ¡ E is dense in X. Note that if E is not closed then its complement X ¡ E contains (properly) the open subset X ¡ E. Proposition 1.6.2. Let X be a Zariski space. A subset E X is constructible if and only if for every closed irreducible subset Y X, the intersection E X Y either contains a nonempty open subset of Y, or is nowhere dense in Y. Proof. See [13], Proposition 1.3, p. 2.

Remark 1.10. Assume we have a subset E of an irreducible Zariski space X with generic point η. In this situation, every nonempty open subset of X is dense. And E is dense if and only if η P E, if and only if E contains an open subset. Assume now that E is constructible: as either E contains an open subset or (aut, in fact) X ¡ E is dense ( nonempty), one concludes that: ¡ If E is constructible in X t η u , then either E contains a nonempty open subset, or X ¡ E does. Equivalently, either E or X ¡ E is a neighborhood of η.

Let us switch to schemes. Here is the aim of the following discussion: suppose we are given a morphism of schemes f : X Ñ S of ﬁnite presentation, and a quasi-coherent OX-module F . First, some notation: if s P S is any point, we denote by Xs the ﬁber X ¢S Spec kpsq and we set b p q 9 Fs F OS k s . It is an OXs -module. Suppose Q is a property

9 ¦ ¡1 j j b ¡1 j : This is nothing but another notation for F F j OX OXs , if Xs Ñ X is the canonical projection. For instance, if X Spec B, S Spec A and ¦ F M , then Xs Spec pB bA kpsqq and Fs j pM q pM bA kpsqq . 1.6. Bertini in characteristic zero 31 of schemes. We then look for conditions under which the locus

E t s P S | Xs has Q u is locally constructible (and hence constructible, when S is an algebraic variety). If one assumes that f is ﬂat, then very often E turns out to be open inside S. The formal discussion starts from the following deﬁnition.

Deﬁnition 1.6.4. Let QpX, F , kq be a relation. It is said to be constructible if the following two conditions hold:

(C1) Assume X is a k-scheme, F is a coherent OX-module and 1 k {k a ﬁeld extension. Then we have

1 1 QpX, F , kq is true ðñ QpXk1 , F bk k , k q is true.

(C2) Let S be an integral noetherian scheme, with generic point ξ. Let u : X Ñ S be a morphism of ﬁnite type and F a coherent ¡1 OX-module. Set, as before, Xs u psq X¢SSpec kpsq and b p q P Fs F OS k s for every point s S. Put

E t s P S | QpXs, Fs, kpsqq is true u .

Then one between E and S ¡ E contains a nonempty open subset (and hence is a neighborhood of ξ).

Proposition 1.6.3. Let Q be constructible, S any scheme and X of ﬁnite presentation over S. Let F be a quasi-coherent ﬁnitely presented OX-module. Then

E t s P S | QpXs, Fs, kpsqq is true u (1.16) is locally constructible. If S is irreducible with generic point ξ then (as we know by Remark 1.10) one between E and S ¡ E is a neighborhood of ξ.

Proof. See [10], Proposition 9.2.3, p. 58.

Remark 1.11. We have a kind of converse of Proposition 1.6.3: if Q is a property satisfying the conclusion of the Proposition (that is, the set in (1.16) is locally constructible), then Q satisfies the condition (C2) in Definition 1.6.4. In fact, take S integral and noetherian as in the definition, and use the fact that in an irreducible noetherian space, a constructible set is either nowhere dense or it contains a nonempty open subset, according to Remark 1.10. Thus if it does not contain an open subset, its complement does. 32 Linear systems

We have the following results.

Proposition 1.6.4. Let X Ñ S be ﬁnitely presented and Q one of the following properties:

¥ geometrically regular;

¥ geometrically normal;

¥ geometrically reduced.

Then the set t s P S | Xs has Q u is locally constructible in S.

Proof. See [10], Corollary 9.9.5, p. 94.

As we announced, it turns out that the flatness property often lets constructible sets become open. In fact, if f : X Ñ S is a morphism of finite presentation, then the set of points x P X at which f is flat is open in X. This observation, together with a strong result (See [10], Theorem 12.1.6, p. 178), allow to prove the following.

Theorem 1.9. Let f : X Ñ S be locally of ﬁnite presentation. Then the set of points x P X where f has one of the following properties: being

1. smooth;

2. normal;

3. reduced;

4. Cohen-Macaulay. is open in X.

Proof. See [10], Corollary 12.1.7, p. 179.

It is time to approach, as promised, the new proof of Bertini’s Theorem in characteristic zero. Let X be an integral and nor- mal10 algebraic variety over a ﬁeld k of characteristic zero. At the very beginning of this section, one took a linear subspace V H0pX, L q, where L is a line bundle on X, and proved that k-linear equivalence in V (that is, the operation of identifying v ¢ to λv for all λ P k ) corresponds to linear equivalence of effective

10 Normality is required to get the bijection v ÞÑ Dpvq, see below. 1.6. Bertini in characteristic zero 33

Cartier divisors on X. In particular, one constructed the divisor of zeros psq psq0 associated to any global section s P V and by the above consideration (in fact, Proposition 1.3.1) one identiﬁed ¢ ¦ pV ¡ t 0 uq{k PV to |D|, where D pv0q is the divisor of some ¦ nonzero v0 P V. This allowed to assert that points in PV parame- terize divisors in |D|. Now, as a notation, one writes Dpvq instead of the old pvq pv0q. As always, Dpvq can be viewed as a hypersurface in X determined by a linear combination of the shape λ0v0 ¤ ¤ ¤ λrvr, if the vi’s constitute a k-basis of V. Now, writing ¢ v for the class of v modulo k , then the bijective correspondence becomes

¦ ...... | p q| PV . D v0

...... v . ... Dpvq. Each divisor E Dpvq in the system has a structure of closed subscheme of X, given by the sheaf of ideals OXp¡Eq OX (See Remark 1.4). More precisely, recall that OXp¡EqpUq fU.OXpUq if E is defined by local data t U, fU u. ¦ ¦ One considered PV as a projective space Pr , and this was the variety of parameters. We now make something a little more subtle. However, the underlying idea is always the same: putting all the divisors Dpvq together in a new algebraic variety Z, which will look like a projective bundle at all points outside the base ¦ locus of PV . In what follows, we denote by PpE q the X-scheme called the projectivization of the coherent OX-module E . It is defined as Proj Sym E , the global Proj of the (graded) symmetric algebra of E . Let us define ¦ ¦ ¦ Z t px, vq P X ¢k PV | x P Dpvq u X ¢k PV PpV bk OXq where the last equality holds because Proj behaves well under ¦ base change (Proposition A.0.5). Of course, we called V the con- ¦ ¦ stant sheaf on Spec k with values in V (here we view V and V as vector bundles over Spec k). ¦ If one shows that Z is a subvariety of X ¢k PV , then one ¦ obtains that q : Z Ñ PV is a family of subvarieties of X with ¦ base PV , where q is the restriction of the canonical projection ¦ ¦ X¢kPV Ñ PV , and where the members of the family are exactly ¡ the closed subschemes Dpvq. In fact, proving q 1pvq Dpvq will require some computation; it is Lemma 1.2. ¦ Proposition 1.6.5. The subset Z is a subvariety of PpV bk OXq.

Proof. Let φ : V bk OX Ñ L be the morphism of (coherent) sheaves that on every open subset U X sends v b f to f.v|U, 34 Linear systems

for every v P V and f P OXpUq: just to ﬁx notations, this means that φpUqpv b fq f.v|U. We can consider the dual ¦ ¦ ¦ φ : L −Ñ V bk OX ¦ and observe that Z Ppcoker φ q. If this is proved and we name ¦ E coker φ , then by exactness of

¦ φ¦ ¦ L −Ñ V bk OX −Ñ E −Ñ 0 ¦ we get the desired inclusion PpE q Z ãÑ PpV bk OXq. We can prove Z PpE q affine-locally, so let us fix an open affine subset U X. We also fix a basis v0,..., vr of V and write vi|U e.fi, for unique fi P OXpUq (that is, we are translating L |U e.OX|U). Our claim is then Z|U PpE q|U, where ¦ ¦ Z|U t px, vq P X ¢k PV | x P U X Dpvq u U ¢k PV and PpE q|U PpE |Uq Proj Sym E pUq. ¦ ¦ ¦ As L pUq e.OXpUq, we have L pUq L pUq e .OXpUq, and regarding the exact sequence ¦ ¦ 0 −Ñ L pUq −Ñ V bk OXpUq −Ñ E pUq −Ñ 0 (1.17) ¦ it is important to know where the generator e goes under (the ¦ ¦ second arrow) φ pUq. In general, if θ P L pUq, then θ is an ¦ OU-morphism L |U Ñ OX|U and φ pUqpθq is the linear map in ¦ ¦ V bkOXpUq pVbkOXpUqq sending vb1 to θpUqpφpUqpvb1qq p qp | q ¦ ¦p qp ¦q θ U v U . In particular, e °goes to the° linear map φ°U e b ¡1 | ¡1 | sending v 1 to e .v U e . i λivi U i λifi (if v i λivi). In fact, simply by using the definition of the dual basis, one sees that ¸ ¦p qp ¦q ¦ b φ U e vi fi. i Thus, looking again at exactness of (1.17), and considering the ¦ ¦ ¦ ¦ ideal pφ pUqpe qq generated by e inside Sym pV bk OXpUqq ¦ V bk OXpUq, one has p ¦ b p qq p ¦q b p q p q Sym V k OX U Sym°V k OX U Sym E U p ¦p qp ¦qq p ¦ b q . φ U e i vi fi ° p q| p q p qr s{p q Now, P E U Proj Sym E U Proj OX U t0,..., tr i fiti ¦ where we have made correspond vi to the indeterminate ti. To conclude, let us choose a k-rational point x P Upkq, and let us compare Zx t px, vq | Dpvq X U Q x u and p qr s p | q p | q ¢ p q k x° t0,..., tr P E U x P E U U Spec k x Proj p p q q . i fi x ti 1.6. Bertini in characteristic zero 35

° P p q Finally, choosing v i λivi with x D v is equivalent to select- ing a point in PpE |Uqx, via the identiﬁcation λi fipxq.

¦ So q : Z Ñ PV is a family of subvarieties of X. The following lemma states it is universal, in the sense that what lies above a ¦ point v P PV is exactly the subscheme Dpvq.

¦ Lemma 1.2. The members of the family q : Z Ñ PV are the subschemes Dpvq. That is, for every nonzero section v P V we have Zv Dpvq as closed subschemes of X.

Proof. The claim can be verified locally on any open affine subset | P U Spec A X. Suppose° that vi U° e.fi with fi A, for ¤ ¤ P 0 i r. Then v i λivi e i λifi, where λi k are coefficients.

¦ Z X pU ¢ PV q Z X pU ¢ Spec kpvqq v k v k ¸ Proj Art0,..., trs{pλitj ¡ λjti, fitiq

Spec A{fU Spec ODpvqpUq.

Remark 1.12. Perhaps now it is a good moment to revisit Remark 1.6. At that time, it could seem obscure why divisors can be in- r ¦ terpreted as preimages of hyperplanes in P under ψ : X 99K PV , ¦ but now, after Lemma 1.2, it should be clear that points of Pr are ¦ exactly the v P PV we are to consider.

¦ Lemma 1.3. If the linear system PV is base-point-free, then the canonical morphism p : Z Ñ X is a projective bundle. In particular, it is smooth.11

Proof. The absence of base points says that φ is surjective, so ¦ ¦ ¦ ¦ coker φ pker φq . Hence Z Ppcoker φ q Pppker φq q, where ¦ pker φq must be locally free of rank r (by looking at the exact sequence 0 Ñ ker φ Ñ V bk OX Ñ L Ñ 0). So Z is a projective bundle over X, via the morphism p.

Theorem 1.10. Let k be a ﬁeld of characteristic zero and X an ¦ integral algebraic variety over k. Let PV be a linear system on X with base locus B. Suppose that X satisﬁes one of the following properties Q: being

1. smooth ( ; Bertini);

2. normal;

11 This is proved in [EGA IV4, p. 62]. 36 Linear systems

3. reduced.

¦ Then there exists a dense open subset Ω PV such that for every rational point v P Ωpkq the corresponding divisor Dpvq (i.e. ¦ the ﬁber of q : Z Ñ PV at v) satisﬁes Q away from B.

¡ Proof. The goal is to find Ω such that the fibers of q : q 1pΩq Ñ Ω satisfy Q. First, by replacing X with the open subset X ¡ B, one may assume the linear system is base-point-free. Indeed, working with X ¡ B we deal with the fibers Dpvq|X¡B instead of Dpvq, ¦ and because X ¡ B is open in X one has that PpV |X¡Bq is open ¦ ¦ in PV . Thus, a nonempty open subset Ω PpV |X¡Bq doing ¦ the job would be open in PV as well, and saying that Dpvq|X¡B satisfies Q is exactly our final claim. So by the absence of base points we know that the projective bundle Z is smooth over its base scheme X, namely, the projection p : Z Ñ X is smooth. This implies that all the fibers of p are smooth, in particular Z itself is smooth. Hence Z satisfies Q.

¥ If Q is the smoothness property, this is the situation: there is ¦ morphism of varieties q : Z Ñ PV over a ﬁeld of character- ¡ istic zero, and Z is smooth. Hence q 1pΩq Ñ Ω is smooth ¦ for some nonempty open subset Ω PV . This is exactly the generic smoothness theorem.

¥ If Q is any of the above properties, then by Proposition 1.6.4 (which we apply because q is of ﬁnite type) the set

¦ E t v P PV | Zv is (geometrically) Q u

is constructible. But now, as char k 0 and Z satisﬁes Q, the generic ﬁber Zη of q is (geometrically) Q. So (see for example our discussion on generic ; general) E contains a nonempty open subset Ω because it contains the generic point η.

This concludes the proof.

Corollary 1. If, in addition to the assumptions in the Theorem, ¦ dim V ¥ 2, then there are inﬁnitely many members in PV satisfying Q.

¦ Proof. Indeed, then PV is ”at least” a P1 over an infinite field, so any nonempty open subset contains infinitely many rational points. 1.6. Bertini in characteristic zero 37

Final Remark. Now k is any ﬁeld and X is an algebraic variety over k. Suppose we are given a k-morphism f : X Ñ Pr (induced by some linear system). In proving Theorem 1.10, and in the preceding discussion, the central object was the family

¦ ¦ q : Z Ñ PV Pr .

We did show that its generic fiber satisfies Q, and then used the constructibility of Q to see that for some nonempty open sub- ¦ ¡ set Ω PV the fibers of q 1pΩq Ñ Ω satisfy Q. We can in fact generalize this procedure. A hyperplane is a particular linear subvariety of the projective space: it has codimension one. But for all 1 ¤ d r we can consider

Grpd, rq t Linear projective L Pr of dimension d u .12

Now we can construct the analog object of our family Z, namely

Z t px, Lq P X ¢k Grpd, rq | fpxq P L u .

Here the condition fpxq P L replaces x P Dpvq in the old Z. This construction is more general because we take linear subvarieties (intersections of ﬁnitely many hyperplanes) instead of hyperplanes. One shows that the projection morphism

qˆ : Z −Ñ Grpd, rq

¡1 ¡1 satisﬁes ZL qˆ pLq f pLq X for every L P Grpd, rq, the analog of our old Zv Dpvq. One might ask: is there an analog of Theorem 1.10, for linear subvarieties of Pr? The answer is yes, so here is the result.

Theorem 1.11. Let k be a ﬁeld, X a k-scheme of ﬁnite type, let 1 ¤ d r and f : X Ñ Pr a k-morphism.

(1) If X is irreducible, thenq ˆ is dominant if and only if dim fpXq ¥ d.

(2) Suppose char k 0 or f not ramified. If X is smooth (resp. geometrically reduced) then so is also the generic fiber of ¡ qˆ . Moreover, if k is infinite, then f 1pLq is smooth (resp. geometrically reduced) for almost all L P Grpd, rq.

(3) If dim fpXq ¥ d 1 and X is geometrically irreducible then the generic ﬁber ofq ˆ is geometrically irreducible.

12 For instance, Grp1, rq t r`s | ` Pr line u PH0pPr, Op1qq¦, and Grpr ¡ 1, rq t rHs | H Pr hyperplane u PH0pPr, Op1qq. 38 Linear systems

Proof. See [13], Theor´ eme` 6.10, p. 89.

Note that the above theorem already answers a question that one could ask: is there a hope to let the strong Bertini’s Theorem become true in positive characteristic? The answer provided in (2) is yes, by requiring that f be unramiﬁed. We will discuss again about this topic later. CHAPTER 2

Applications and Pathologies

In this chapter we make some remarks and examples of ap- plication of Bertini’s Theorem: most of all, we will deal with the version holding in any characteristic, and we will point out some basic consequences, like the existence of smooth hypersurfaces of any degree contained in a smooth variety. For example, any smooth surface will contain (many) smooth curves. A few explicit computations will be provided, and we will also focus on bad situations, namely when Bertini cannot hold, because of some missing hypothesis. We will be able to explain the failure of the strong Bertini in characteristic zero, with an explicit example in characteristic 2.

2.1 First positive results

Before beginning with examples, let us describe what we mean n by the dual variety of an algebraic variety X. Assume X Pk Proj krx0,..., xns is a smooth algebraic variety over a field k. We define the dual variety, as a set, to be ¦ t | P u X TX,x x X . ¦ If X is not smooth, then X is defined to be the Zariski closure p q Ñ n¦ ÞÑ of δ Xsm where δ : Xsm Pk is the morphism x TX,x and ¦ Xsm is the set of smooth points of X. In any case, X is a closed n¦ subvariety of Pk , hence itself a projective variety; it has the reduced subscheme structure but it might neither be irreducible nor of pure dimension.

39 40 Applications and Pathologies

2 ¦ 2¦ For example, let X be a curve in Pk. Then X Pk is the set of tangent lines through the (smooth) points of X. It need not be smooth even if X is. As an example, consider a smooth, i.e. p q 2 ¦ non degenerate conic C V f in Pk. Then C C . Indeed, ¦ by looking at the gradient map p ÞÑ ∇pf one sees that C is also a conic (unless char k 2). But this is a very special case: in general, the dual curve of a degree d curve has degree dpd ¡ 1q, so it is not isomorphic to the original curve.

Example 2.1.1.( More on the Space of Conics) Let k k be a ﬁeld of characteristic 2. Recall that on P2 we have the complete linear system of conics D PH0pP2, Op2qq P5. Let us verify directly that Bertini’s Theorem holds. Let us choose, for instance, the (smooth non degenerate) conic X given by

p q 2 2 2 f x0, x1, x2 x0 x1 x2

¦ and let us see that, for a sufﬁciently general H P P2 , the section H X X remains smooth. So let us take a hyperplane Ha : a0x0 p q P 2 a1x1 a2x2 0, corresponding to the point a a0, a1, a2 P . We may assume a1 0. We are interested in checking that the set ℵ of hyperplanes such that their intersection with X is smooth is 2¦ a dense open subset of P . Let us call Ya Ha X X. It is smooth at p pα, β, γq P Ya if and only if the Jacobian matrix ¢ ¢ Bf Bf Bf Bx Bx Bx 2α 2β 2γ Jp 0 1 2 ppq a0 a1 a2 a0 a1 a2

1 has maximal rank, i.e. rank Jp 2. Assuming β 0, this happens if and only if $ $ ¡ { { &2αa1 2βa0 0 &α β a0 a1 ¡ ðñ { { %2αa2 2γa0 0 %α γ a0 a2 ¡ { { 2βa2 2γa1 0 γ β a2 a1 where we have assumed that γ 0 (indeed α γ 0 would lead to a contradiction). So p pα{β, 1, γ{βq is a smooth point of Ya p q if and only if p a0, a1, a2 a. We deduce that Ya is smooth precisely when a R Ya, which is equivalent to a R X. So we ﬁnd

2¦ ¦ ℵ t Ha | Ya is smooth u P ¡ X which of course is dense.

1 Vocabulary: the good property of having smooth intersection with X is sometimes referred to as ”H intersects X transversally”. 2.1. First positive results 41

n Example 2.1.2. More generally, let X Pk be a smooth hypersurface, so that dim X n ¡ 1, which is the dimension of any P n¦ hyperplane H Pk . Consider, as above, ℵ t Hyperplanes H such that H X X is smooth u .

Recall (perhaps, by looking at the proof of Bertini’s Theorem) that P X x H X is a smooth point if and only if TX,x H, if and only R ¦ if TX,x H, which is equivalent to H X . The ﬁrst equivalence comes from the fact that, as X is a smooth hypersurface, TX,x is a hyperplane, that is, a hypersurface given by a linear form, hence of dimension n ¡ 1 dim X, so being contained means being equal. We conclude that n¦ ¡ ¦ ℵ Pk X ¦ and ℵ is not empty because dim X dim X n.

n Example 2.1.3. We prove that in Pk there exists ”many” smooth hypersurfaces of any ﬁxed degree. Let us consider the smooth n variety X Pk , over any algebraically closed ﬁeld k. Consider the Veronese d-uple embedding

... .. n ...... m vd : Pk .... Pk ¨ n d ¡ where m d 1. Assume that x0,..., xn are the coordinates n m on Pk and T0,..., Tm are those on Pk . The morphism vd is given, locally, by a morphism θ : krT0,..., Tms Ñ krx0,..., xns sending Ti to some monomial of degree d in the xj’s. Thus if gpT0,..., Tmq is any homogeneous polynomial of degree r and V is the hyper- m X n surface it defines in Pk , then V Pk is the hypersurface given by the polynomial gpθpT0q,..., θpTmqq, hence of degree dr. Hence, if m X n H is any hyperplane in Pk , the variety H Pk is a smooth hyper- n surface of degree d inside Pk . There exists one for every positive integer d. What Bertini says is that the set of these smooth hypersurfaces is a dense open subset of the complete linear system | p q| n O d on Pk . Example 2.1.4. As a last example, we prove that Bertini’s The- orem continues to hold if X has finitely many singular points, s t u say X p1,..., pt . The set of transversal hyperplanes U t | P u n¦ H H TX,x for all x X Pk is a nonempty open subset, because it is obtained as the intersection of finitely many (just by assumption) open subsets of the dual projective space. Consider ¤t 1 s U t H | H X X H u t H | pi P H u . i1 42 Applications and Pathologies

t | P u n¦ 1 Every H pi H is a hyperplane in Pk , so U is a ﬁnite union n¦ ¡ 1 of hyperplanes, and hence a closed subset of Pk . Thus U U is still open (thus dense), and every H0 inside it satisﬁes H0 TX,x s for any point x P X and H0 X X H, thus H0 X X is smooth, and Bertini’s statement then follows. If, instead, dim Xs ¥ 1, then by the Dimension Theorem we get H X Xs H for every hyperplane H, thus H X X is never a smooth scheme.

To conclude this section, we give the following result. It will be useful later.

Theorem 2.1. Let X be an integral, normal projective variety of dimension at least 2. Let Y X be a closed subset of codimension one, which is the support of an effective ample divisor. Then Y is connected.

Proof. See [14], III, Corollary 7.9, p. 244.

A consequence of Theorem 2.1 is that when dim X ¥ 2 in Bertini’s Theorem (any characteristic), the hyperplane sections Y H X X are in fact connected, and since they are smooth by Bertini, they are irreducible. In fact, a hyperplane section is by deﬁnition an effective (it corresponds to the pullback of Op1q) am- ãÑ n ple divisor on X, once one has ﬁxed an embedding X Pk .

2.2 Something goes wrong: Frobenius

Let X be an algebraic variety. When char k p ¡ 0, strange things can happen. In fact, there are two possibilities:

¥ The field k is infinite. Then one always finds a hyperplane, defined over k, cutting X transversally. If k, in addition, is algebraically closed, then the (weak) Bertini Theorem applies and we know that there are many good hyperplanes, intersecting a smooth X transversally. But strong-Bertini might fail. In fact, even for k k, weak-Bertini might fail as well, n if one does not embed X in Pk in the right way (that is, via a closed immersion).

¥ If k is ﬁnite, we are in a trouble. All the ﬁnitely many hyperplanes might fail to cut X transversally, even if X is smooth. So even weak-Bertini might be false.

Zariski gave the following example: consider, for any (perfect) ﬁeld k of positive characteristic, the pencil of curves inside P2 2.3. Smooth Complete Intersection 43

p p p generated by x and y . Then every curve is of the form Cλ : x { λyp 0, for λ ranging P1. But then 0 xp λyp px λ1 pyqp says that every point in Cλ is a p-fold point, in particular it is singular. This is an example of a linear system on P2 with (lots of) singularities outside the base locus (which is one point). So we proved the possible failure of strong-Bertini, and the above example applies both to finite and infinite (perfect) fields. We now illustrate the possible failure of weak-Bertini over any k k of characteristic p ¡ 0. Consider any smooth X p q n p q V f1,..., fr P where fi x0,..., xn are homogeneous. We show that for a particular embedding of X in Pn (via Frobenius) there is no smooth hyperplane section. More precisely, we consider the composition of a fixed closed immersion with the Frobe- nius morphism (see Appendix for the construction of the relative Frobenius F FX{k). More precisely, call f the composition

F p q X −Ñ X p X ãÑ Pn.

The point is that f is not a closed immersion (F is not). We show n¦ that there is no hyperplane V phq H P P such that H X X ¡ f 1pHq is a smooth variety. Let H be any hyperplane, and let us p q r s verify our claim locally, say on U D x0 Spec k u1,..., un , where uj denotes the afﬁne coordinate xj{x0. The corestriction | f U, still denoted by f, induces

7 f ... r s ...... r s{p q k u1,..., un k u1,..., un f˜1,..., f˜r ...... ¡ n pUq p 1p qq OPk OX f U p q p q where f˜i u1,..., un fi 1, u1,..., un . Thus

f¡1pU X Hq f¡1pVph˜ qq Vpf7ph˜ qq Vph˜ pq r s{p pq Spec k u1,..., un f˜1,..., f˜r, h˜ , which is not even reduced.

2.3 Smooth Complete Intersection

Throughout this section, one refers to weak-Bertini (in any characteristic). So k is any algebraically closed ﬁeld. The proofs of the main assertions in Commutative Algebra can all be found in [3]. The main topic will be complete intersection in projective space: one will be able to apply Bertini’s Theorem to show that for 44 Applications and Pathologies

n every r n there exists a codimension r subvariety Y Pk which is a smooth irreducible complete intersection (of hypersurfaces of prescribed degree). n Deﬁnition 2.3.1. A closed subscheme Y Pk is said to be a (global) complete intersection if its homogeneous ideal IpYq r s p nq k x0,..., xn can be generated by r codim Y, Pk elements.

Remark 2.1. We take as a deﬁnition IpYq Γ¦pIYq, for every Y projective over Spec A. In fact, Γ¦pIYq is the largest ideal deﬁning n Y inside PA. The aim of this section is resumable in three points. 1. We will show this result:

n Theorem 2.2. A closed subscheme Y Pk of codimension r is a complete intersection if and only if there are r n X ¤ ¤ ¤ X hypersurfaces H1,..., Hr in Pk such that Y H1 Hr, scheme-theoretically. On the level of sheaves, this means ¤ ¤ ¤ that IY IH1 IHr .

n ¡ 2. If Y Pk is a complete intersection of dimension 0 and Y is normal, then it is projectively normal; moreover, for ¥ 0p n p qq Ñ 0p p qq all ` 0, the natural map H Pk , O ` H Y, OY ` is surjective. When ` 0 this says that Y is connected. ¥ 3. For any r n choose d1,..., dr 1 to be arbitrary integers. n Then there exist smooth hypersurfaces Hi Pk , of degree X ¤ ¤ ¤ X di, such that the scheme Y : H1 Hr is irreducible, n nonsingular and of codimension r in Pk . One can already show one direction of Theorem 2.2, the ’only if’ one. Indeed, if Y is a complete intersection then IpYq can be p q p q generated by r elements, say I Y f1,..., fr where r is the n X ¤ ¤ ¤ X codimension of Y in Pk . Hence Y H1 Hr, scheme- theoretically, where Hi V pfiq for i 1, . . . , r. Another way to see the ’only if’ part is by considering sheaves: we have Γ¦pIYq p q p q p q I Y f1,..., fr and I Y IY, so ¡¸ © ¸ ¸ p q p q p q IY f1,..., fr fi fi IHi

as the ideal sheaf of Hi V pfiq is pfiq . The next efforts are devoted to show the converse.

§ Tool I: Some Primary Decomposition. The following holds for every commutative unitary ring A, if one replaces ’ideal’ by 2.3. Smooth Complete Intersection 45

’decomposable ideal’. But we assume that A is a noetherian ring, thus every ideal a A admits a primary decomposition, i.e. one can write £m a qi (2.1) i1 where qi are primary ideals. Saying that q is primary means that q A and, whenever xy P q and x R q, one has yr P q for some r ¡ 0. (Equivalently, A{q 0 and every zero divisor of A{q is nilpotent.) Note that the radical of a primary ideal is necessarily prime, but the power? of a prime need not be primary; we say that q is p-primary if p q.

Remark 2.2. Because a ﬁnite intersection of p-primary ideals is p- primary, one can? always ﬁnd a primary decomposition in which all the radicals qi are distinct; if, in addition, for every i 1, . . . , m one has qi ji qj, then the decomposition is said to be minimal (we always have one such, of course).

Theorem? 2.3. Given a minimal decomposition like in (2.1), with pi qi, one has the following identiﬁcation: t u t u p { q p1, p2,..., pm Primes associated to a AssA A a .

Proof. See [3], Theorem 4.5, p. 52.

Deﬁnition 2.3.2. A prime ideal p A is said to be an associated prime for a in case p Ann px aq for some x P A. The minimal elements in AssApA{aq are called isolated/minimal primes, while all others are called embedded primes.

Example 2.3.1. Consider the ideal a px2, xyq in A krx, ys. X 2 p q p q Then we obtain a as p1 p2, where p1 x q1 and p2 x, y . 2 The ideal q2 p2 is p2-primary because it is a power of the maxi- X mal ideal p2. Hence q1 q2 is a (minimal) primary decomposition p { q t u for a and AssA A a p1, p2 , where p1 is isolated and p2 is embedded (in this case it is easy to see, as p1 p2). The geometric meaning of the terminology isolated/embedded prime is the following: consider the closed subvariety X of A2 deﬁned by the ideal a. Then, in this example, X Vpxq is the y-axis, which is irreducible and thus corresponds to the unique irreducible component (that is, minimal prime in the usual sense) given by pxq. Instead, the embedded ideal px, yq corresponds to the origin, clearly lying in X. So we have to keep in mind that an isolated/minimal prime corresponds, ideally, to an irreducible component of the variety deﬁned by a, and an embedded prime corresponds to a variety 46 Applications and Pathologies embedded in some irreducible component. Algebraically, an isolated prime p is characterized by the properties

p a, and for every b a one has p b. ² As AssApA{aq t Isolated primes u t Embedded primes u, an embedded prime is characterized by

p a, and for some b a one has p b.

Finally, because any prime ideal p a also contains an isolated prime associated to a, we have that

t Isolated primes associated to a u t Minimal primes above a u and we are very happy with this, because we recover the usual correspondence between minimal primes above a and irreducible components of the variety Spec A{a.

§ Tool II: Cohen-Macaulay rings and unmixed ideals. Let A be a ring and M an A-module. We say that a finite sequence t u x1,..., xr A is regular for M if x1 is not a zero divisor in M (that is, there exists m 0 such that x1.m 0) and for all i 2, . . . , r the element xi is not a zero divisor in M{aiM, where p q ai x1,..., xi¡1 . The latter condition means that there exists a nonzero m aiM, namely m R aiM, such that xi.m P aiM. Definition 2.3.3. Let pA, mq be a local ring and M an A-module. One writes depthA M for the depth of M, the maximum length t u of a regular sequence x1,..., xr m for M. If A is noetherian (and local) one defines its depth in the same way, and we call A a Cohen-Macaulay local ring if depth A dim A. A noetherian (not necessarily local) ring A is said to be Cohen-Macaulay (and we write CM) in case every localization Ap at a prime (equivalently, maximal) ideal of A is a CM local ring. This suggests that, on the scheme side, the CM property will be stalk-local, and in fact it is so.

Deﬁnition 2.3.4. Let A be a noetherian ring and a an ideal with p { q t u associated primes AssA A a p1,..., pm . Then a is said to be unmixed if it shares its height with all of its associated primes, i.e. ht a ht pi for all i 1, . . . , m. Now let us consider the following property.

p:q Every ideal a A of height r and generated by r elements is unmixed. 2.3. Smooth Complete Intersection 47

One has the following results.

Theorem 2.4. A noetherian ring A is CM if and only if p:q holds in A.

Theorem 2.5. A noetherian ring A is CM if and only if the polynomial ring Arxs is CM. In particular, if A is CM (e.g. a ﬁeld) r s then the ring A x1,..., xn is also CM.

n Proof. (of Theorem 2.2) Recall that we have some Y Pk of codi- X ¤ ¤ ¤ X mension r. We still have to show that if Y H1 Hr scheme- theoretically, where Hi V pfiq are hypersurfaces, then Y is a p q complete intersection. We have to show that, if a f1,..., fr and b IpYq, then a b. By Theorem 2.5, the ring S krx0,..., xns is CM, so p:q holds in S by Theorem 2.4. This implies that the p nq ideal a S, which is of height r because codim Y, Pk r, is unmixed. Hence every p P AssSpS{aq has height r. Now, we are in a situation where a and b deﬁne the same projective k-scheme r ¤ ¤ ¤ r and in fact b IY IH1 IHr a (this translates the scheme-theoretic equality), so that the fi’s all belong to b and hence a b. For every j 0, 1, . . . , n we have apxjq bpxjq and Nj this implies that xj b a for some positive integer Nj. Indeed, for every d ¡ 0 and for every homogeneous element α P bd there 1 P { d 1{ d exists some α ad such that α xj α xj . This implies that dp ¡ 1q d d 1 P xj α α 0, that is, xj α xj α a. This holds for every d " Nj and for every α, thus there is some Nj 0 such that xj b a. By setting N maxj Nj one gets that N S b a b. Now we denote by a bar the reduction mod a. Let us notice that N b b{a is an ideal of S{a satisfying S b 0. Let us suppose, by contradiction, that b is nonzero. Then there exists some nonzero N element α P b and by the above we have S α 0, which says N S t x P S | xα P a u Ann pα aq p P AssSpS{aq.

As S is prime, we have in fact that S is contained in the minimal prime p. But as it is maximal we must have S p, which is a contradiction because n 1 ht S ¡ r ht a. Thus we conclude that a b. So Theorem 2.2 is proved and the ﬁrst part of our purpose is complete. Let us pass to part 2, about projectively normal varieties. 48 Applications and Pathologies

n Deﬁnition 2.3.5. Let A be a ring. A projective A-scheme Y PA is said to be projectively normal if its homogeneous coordinate ring SpYq Arx0,..., xns{IpYq is normal (integrally closed), where IpYq Γ¦pIYq. n Let us show that a normal complete intersection Y Pk of strictly positive dimension must be projectively normal. We will p q n 1 use the following result, applied to the afﬁne cone C Y Ak .

Theorem 2.6. Let X be a smooth variety over an algebraically closed ﬁeld k and let Y X be a local complete intersection subvariety. Then Y (is CM and) is normal if and only if it is regular in codimension one.

Proof. See [14], II, Proposition 8.23, p. 186.

n Let us call r the codimension of Y in Pk and, using our hy- X ¤ ¤ ¤ X p q pothesis, let us write Y H1 Hr, remembering that I Y p q n 1 ¡ t u Ñ n f1,..., fr . By considering the projection θ : Ak 0 Pk then the afﬁne cone C CpYq over Y is deﬁned to be

¡1p q Y t u p q n 1 C θ Y 0 Spec S Y Ak .

Here SpYq is krx0,..., xns{IpYq, the homogeneous coordinate ring of Y. It coincides with the afﬁne coordinate ring of C. Thus our goal is to prove that C is normal. If one shows that it is a local n 1 complete intersection in Ak then one is left to prove that it is regular in codimension one. In fact, C is even a global complete p q p q intersection, because its ideal I Y f1,..., fr is generated by p nq p n 1q r elements, where codim Y, Pk r codim C, Ak because dim C dim Y 1. It remains to prove that C is regular in codimension one. Of course Y has this property because it is normal by assumption, but we will not use this, we will just use that Y is normal. Let us start by noticing that if we corestrict θ to, say, D px0q we obtain | p q D x0 θ : Dpx0q −Ñ D px0q.

This corresponds to the ring homomorphism x1 xn 1 x1 xn 1 k ,..., −Ñ k x0,..., xn, k ,..., x0, . x0 x0 x0 x0 x0 x0 p q ¡ t u Now if we write yi for the class xi mod I Y then C 0 ¡1p q p q p q θ Y i D yi and the corestriction of θ to D yi Y is Dpyiq Ñ D pyiq, and this holds for all i 0, . . . , n. By the above, we remark that C ¡ t 0 u is, locally on each Dpyiq, the product 2.3. Smooth Complete Intersection 49

of the afﬁne schemes D pyiq ¢k Gm, where we identify Gm to ¡ Spec krt, t 1s. Hence

OpDpyiqq OpD pyiq ¢k Gmq OpD pyiqq bk OpGmq ¡1 ¡1 OpD pyiqq bk krt, t s OpD pyiqqrt, t s and the latter is a normal ring as, in general, if A is normal then Arts is normal, and the same holds for any localization of A. Here of course we used that each OpD pyiqq is normal. So we conclude that C ¡ t 0 u is normal. At the origin, we have that

¡ dim OC,0 dim C dim Y 1 1 so we get that C is regular in codimension one, as claimed. To conclude part 2, we need the following result (which holds, more generally, if one replaces the ﬁeld k with any ring A):

n Theorem 2.7. A projective k-scheme Y Pk is projectively normal if and only if it is normal and, in addition, the natural 0p n p qq Ñ 0p p qq ¥ map H Pk , O ` H Y, OY ` is surjective, for every ` 0.

Proof. See [Hartshorne, II, Ex. 5.14(d)]

So in particular we have that for a closed projectively normal n 0p n p qq Ñ 0p p qq subvariety Y Pk all the morphisms H Pk , O ` H Y, OY ` 0p n q are surjective. When ` 0 we have that H Pk , O k surjects 0 onto H pY, OYq OYpYq, hence dim OYpYq ¤ 1, which says exactly that Y is connected. Indeed, if it were not connected, one could decompose OYpYq as a direct sum of copies of k. Part 2 is now complete. To prove the third point of our initial goal, we will iterate, in some sense, the reasoning we did in Example 2.1.3. So now we are assuming that r n and we have r positive integers d1,..., dr. We want to construct a smooth irreducible complete intersection n Y Pk given by the intersection of r hypersurfaces of the¨ given n di degrees. To start with, we deﬁne the integer mi ¡ 1. In di Example 2.1.3 we saw that if we consider the d1-uple embedding n ãÑ m1 v1 : Pk Pk then (by Bertini) there is (in fact, there are many) m1 a hyperplane in Pk that pulls back to a smooth hypersurface n p q m2 H1 Pk of degree d1. Now, v2 H1 Pk is a smooth variety, m2 and again by Bertini there is a hyperplane H Pk such that X p q ¡ H v2 H1 is smooth of dimension dim H1 1. The preimage of X this intersection under v2 is some H1 H2, where H2 is a smooth hypersurface of degree d2. Note that Bertini also ensures that 50 Applications and Pathologies the dimension drops by exactly one at each step. One can go on X ¤ ¤ ¤ X like this r times, and at the end Y : H1 Hr is smooth of n codimension r in Pk . So it is a normal complete intersection of positive dimension (because r n) and by part 2 we get that Y is connected; as it is smooth, it is also irreducible, as claimed.

Summary: Thanks to Bertini, we have just proved that for every r n there exists a codimension r subva- n riety Y Pk which is a smooth irreducible complete intersection, given by intersecting r hypersurfaces of prescribed degree.

2.4 Moving singularities: failure in char 2

Throughout this section, one refers to the (stronger) version of Bertini in characteristic zero (a general member of a linear system is smooth away from the base locus), and one shows that it may fail in positive characteristic. In the following example one deals with a linear system of cubics of dimension 2 (a net), all of whose members are singular, with singular points (inside and) outside the base locus. Let k F2 be an algebraically closed ﬁeld of characteristic 2. Let us name T the scheme P2 t P ,..., P u (see Figure 2.1 below). One F2 1 7 deﬁnes D to be the linear system (over k) of all the cubics passing through the Pi’s.

Figure 2.1: The projective plane over F2.

Final goal. First, it will be shown that D is a net with base locus equal to T, and the associated morphism 2 ¡ Ñ 2 Pk T Pk is inseparable of degree 2. After that, we will see that all the curves C P D are singular: either C consists of three lines passing through (exactly) one of the Pi’s, or C is an irreducible cuspidal cubic curve 2.4. Moving singularities: failure in char 2 51

with cusp S R T. Furthermore, associating to C P D its (unique) singular point gives a 1-1 correspondence 2 between D and Pk.

We recall the following:

Deﬁnition 2.4.1. A morphism of, say, integral schemes f : X Ñ Y is said to be purely inseparable if it is injective and if for every x P X, the extension of residue ﬁelds kpfpxqq ãÑ kpxq is purely inseparable. It is called inseparable if KpYq ãÑ KpXq is inseparable (i.e. we drop the injectivity2 and we just have to check at the generic point).

Let us construct explicitly D. Before that, let us list the points in T: p q p q p q P1 1, 0, 0 P2 0, 1, 0 P3 0, 0, 1 p q p q p q p q P4 1, 1, 0 P5 1, 0, 1 P6 0, 1, 1 P7 1, 1, 1 .

2 If x, y, z are homogeneous coordinates on Pk, the generic cubic 2 C Pk is given by the vanishing of the polynomial 3 2 2 3 a1x a2x y a3x z a4y a5xyz 2 2 3 2 2 a6xz a7xy a8z a9y z a10yz . Imposing that C passes through P1, P2, P3 forces a1 a4 a8 0. Passing through P4 gives a2 a7 0, which means a2 a7. Similarly, for P5 we get a3 a6, while P6 gives a9 a10 and passing through P7 forces a5 0. Thus, after renaming the survived coefﬁcients, we get that the general member of D is

fpx, y, zq apx2y xy2q bpx2z xz2q cpy2z yz2q (2.2) so that D is a two-dimensional linear system, generated by the three cubics appearing in the expression of f. One easily sees that X X BD C1 C2 C3 T, where Ci are the generators of D. Therefore there is a morphism

... 2 ¡ ...... 2 ψ : Pk T Pk

. .. p q ...... 2 2 2 2 2 2 x, y, z . . px y xy , x z xz , y z yz q

F 2 The reason is that we want that a composition Z Ñ X Ñ Xppq, with F the relative Frobenius, be an inseparable morphism, but it has no hope to be injective whenever the ﬁrst arrow is not injective. 52 Applications and Pathologies and one has to show it is inseparable of degree 2: it is enough to p q 2 do this locally, for example in the open subset U D z Ak, where z 1. The coordinates on U are s x{z and t y{z. The restriction of ψ becomes

2 2 ...... 2 ppP ¡ Tq X Uq ¡ V py yq ... A k k ¨ . 2 2 2 ...... x y xy x x px y q . ... , , 1 y2 y , y2 y

More precisely, we have

x2y xy2 x2 yx , y2 y y 1 thus the induced morphism on function ﬁelds is deﬁned as follows: ...... j : kps, tq ...... kpx, yq

. .. x y ...... s . . y 1 x

. .. x x ...... 1 t . . y 1 y One has to show that kps, tq kpjpsq, jptqq : K kpx, yq : L is an inseparable extension of degree 2. First, let us notice that

x 1 x y yjptq jpsq x x. y 1

By replacing this value of x in, say, jpsq, we get

x y 0 jpsq jpsq jpsq x y 1 p qp q 2 p q2 p q 2 p q p q p q2 p q j s y 1 y j t yj ts y j t yj ts j s yj s y 1 whence the relation

y2jptqpjptq 1q jpsqpjpsq 1q 0 ; y2 c 0

p qp p q q{ p qp p q q P with c j s j s 1 j t j t? 1 K. So the minimal polynomial of y over K is u2 c pu cq2, which is inseparable of degree 2.

Let us pass to the second part of the ﬁnal goal, regarding singularities in D. First, let us show that every curve in D is singular, 2.5. Kleiman-Bertini Theorem 53 by building the system $ ' Bf ' ay2 bz2 0 $? ? 'Bx ' a y b z ' ' &' &' Bf ? ? 2 2 'B ax cz 0 ; ' a x c z ' y ' ' %'? ? ' %' Bf b x c y bx2 cy2 0 Bz ? ? ? which gives the singular point S p c, b, aq. One notes that every Pi is singular for exactly one cubic in D, obtained by letting a, b, c lie in F2. For example, P1 is singular on the cubic curve y2z yz2 which, in addition, is a union of three lines. More precisely:

Points in F2 Coefficients Equation

p q P1 a b 0, c 1 yz y z P2 a c 0, b 1 xzpx zq P3 b c 0, a 1 xypx yq p qp q P4 a 0, b c 1 z x y x y z P5 b 0, a c 1 ypx zqpx y zq P6 c 0, a b 1 xpy zqpx y zq P7 a b c 1 px yqpx zqpy zq During the computations, one easily notices that the unique way of getting a union of three lines is in fact by choosing a, b, c in F2. So the 7 above equations classify all cubics with a singular? ? point? in T, and any other cubic will be singular in S p c, b, aq but will never split into three lines. Finally, two different triples of coefﬁcients will necessarily give rise to different singular points, so one concludes that the association

...... 2 D .. Pk

...... C . ... SC is a bijection, and the singularities in D move all over.

2.5 Kleiman-Bertini Theorem

The aim of this section is to explain a generalization of Bertini’s Theorem due to Kleiman (see [16] for the result in its original form, or: [14], III, Theorem 10.8; or [12], Theorem 17.22 on page 219). 54 Applications and Pathologies

From now on, the base ﬁeld k is assumed to be algebraically closed, and all schemes considered are integral k-varieties. A homogeneous space is a couple pX, Gq where X is a variety and G is a group variety acting on X, in such a way that the group Gpkq acts transitively on Xpkq. Let us start by giving the friendly statement presented in [12], even if it will be generalized later:

Theorem 2.8 (Kleiman-Bertini, special case). Let pX, Gq be a homogeneous space. Let Y X be a smooth subvariety and g : Z Ñ X a scheme over X. For s P G, deﬁne Vs sY ¢X Z Z, the pullback of the diagram

Z ...... g ...... XsY where sY is isomorphic to Y but is included in X in a different way, namely the embedding sY Ñ X sends y ÞÑ sy. Then, for a general s, the singularities of Vs are all contained in the singular locus of Z, that is pVsqsing Vs X Zsing.

As a consequence, consider the case where g is an inclusion. So what we have is two subvarieties Y, Z of X and one of them is regular. But if both are regular, then sY X Z is regular for general s P G. In words: the general translate of a smooth subvariety of a homogeneous space meets transversally any other smooth subvariety. Kleiman starts its paper by proving the following:

Lemma 2.1. Given a diagram of integral schemes

W Z ...... p ...... q g ...... S X the following facts are true:

(1) if q is ﬂat then there is a dense open subset U S such ¡1 that for every s P U, either p psq ¢X Z is empty, or it is ¡ equidimensional of dimension dim p 1psq dim Z ¡ dim X. ¡1 (Here, of course, p psq W ¢S Spec kpsq W.) 2.5. Kleiman-Bertini Theorem 55

¡1 (2) If q is smooth and Z is regular, then p pξq ¢X Z is regular, ¡1 where ξ is the generic point of S. If char k 0 then p psq¢X Z is regular for s ranging a dense open subset of S. Essentially, everything that is inside Kleiman’s paper follows from the above lemma. First of all, the following theorem (gener- alizing slightly Theorem 2.8). Recall that if s P G then for every X-scheme f : Y Ñ X the translate sY is (isomorphic to) Y, viewed as an X-scheme via the map y ÞÑ sfpyq.

Theorem 2.9 (Kleiman-Bertini). Let pX, Gq be a homogeneous f g space. Consider Y −Ñ X Ð− Z two maps of integral k-varieties (k any algebraically closed ﬁeld). (1) There exists a dense open subset U G such that for every s P U, either sY ¢X Z is empty or it is equidimensional of dimension dim Y dim Z ¡ dim X.

(2) If char k 0 and both Y, Z are regular then there is a dense open subset U G such that for every s P U the variety sY ¢X Z is regular.

Proof. The strategy is to recover the hypothesis of the Lemma: Kleiman shows that in the diagram G ¢ Y Z ...... p ...... q g ...... G X the morphism q sending ps, yq ÞÑ sfpyq is flat (so (1) follows) and in fact - thanks to regularity of Y - it is smooth (so (2) follows, ¡ as p 1psq t s u ¢ Y sY). The smoothness of q is proved by showing that, in addition to flatness, its fibers are geometrically regular and equidimensional.

As a particular (and important) situation, consider when both Y, Z X are subvarieties of a homogeneous space pX, Gq. Then the pullback along X becomes the scheme-theoretic intersection, and we get that: (1) for general s P G, the scheme sY X Z is purely of dimension dim Y dim Z ¡ dim X.

(2) If char k 0 and both Y, Z are regular then for general s P G, the scheme sY X Z is smooth. 56 Applications and Pathologies

As another consequence, there is our old friend.

Corollary 2 (Bertini). Let k be algebraically closed of characteristic zero. Suppose Pr is a linear system on an integral scheme Z. Then a general element in the system is smooth away from the base locus and the singularities of Z.

Proof. Here the homogeneous space X is pPr, PGLpr, kqq. By replacing Z by Z ¡ pB Y Zsingq (the base locus B and the singular locus Zsing being both closed in Z), we may assume that Z is smooth and the system is base-point-free. Then we have a morphism ψ : Z Ñ Pr and by construction the elements of the linear system are the preimages of the hyperplanes H Pr under ψ, ¡1p q ¢ that is: DH ψ H H Pr Z is the divisor corresponding to H. The two arrows appearing in Kleiman-Bertini Theorem are then H ãÑ Pr Ð Z where H is a ﬁxed hyperplane. Because the action of PGLpr, kq is transitive on hyperplanes, we can conclude that for general H the element DH is smooth.

Remark 2.3. Every result that has been deduced in characteristic zero up to now (look at all the items labelled (2)) is still true if one replaces ’regular’ (or smooth) by: reduced, normal, Cohen- Macaulay.

One might ask: what is the hurdle that one has to clear to let Bertini’s Theorem (Corollary 2) become true over a ﬁeld of positive characteristic?

Answer: Ramiﬁcation. In fact, inseparability.

Just to be clear:

Deﬁnition 2.5.1. A morphism f : S Ñ T of k-varieties is said to be unramified if for all s P S, and letting t fpsq, one has that ms mtOs (via Ot Ñ Os) and the extension kptq kpsq is ﬁnite and separable.

Theorem 2.10 (Bertini for k k and char k p). Let Z be a regular integral scheme and Pr a base-point-free linear system on Z. Then a general element of the system is regular if Z separates inﬁnitely near points, that is, if the associated morphism ψ : Z Ñ Pr is unramiﬁed.

Proof. See [16], Corollary 12, p. 296. 2.6. Abelian varieties are quotients of jacobians 57

Remark 2.4. In the previous section, when we dealt with the linear system of cubics passing through P2 , we proved that the mor- F2 phism P2 ¡ T Ñ P2 is inseparable by looking at the generic point of the projective plane. So in particular it is not unramiﬁed. This fact agrees with the above theorem.

2.6 Abelian varieties are quotients of jacobians

In this section the reference is [1]. The motivation for this short discussion is just that we have a chance to use Bertini’s Theorem in order to prove something nontrivial, and moreover the proof here presented is an example of how Bertini can be useful in ”reduction steps”. The aim is showing:

Theorem 2.11. Any abelian variety A over an inﬁnite ﬁeld k is a quotient of a jacobian variety.

An abelian variety is a geometrically integral and projective algebraic group. Note that if dim A 1 then A is an elliptic curve, thus it is isomorphic to its own jacobian and the above statement is satisfied. Let C be a nonsingular projective curve of genus g ¡ 0 and let J be its jacobian. We first describe the universal property of an important arrow mapping to the jacobian. We will need it during the proof of Theorem 2.11. So, the situation is the following. The map F : Cpkq ¢ Cpkq Ñ Jpkq sending pQ, Pq ÞÑ rQs ¡ rPs happens to be defined over k even if C has no rational points. To prove this, one has to show that for every Galois extension of k and for every σ in the Galois group, one has σF F. This k- morphism is universal in the following sense: for any map θ : C ¢ C Ñ A to an abelian k-variety A, satisfying θp∆q 0, there is a unique homomorphism ψ : J Ñ A such that θ ψ ¥ F. We will deal with a smooth curve C A so our θ will be the map pQ, Pq ÞÑ Q ¡ P. Lemma 2.2. Let X be a nonsingular projective k-variety of dimen- ãÑ n sion at least 2, and let us fix a closed embedding X Pk . Let Z X be a hyperplane section relative to this embedding and V a nonsingular variety with a finite morphism π : V Ñ X. Then ¡ π 1pZq is geometrically connected. Proof. One may assume k k, because closed immersions, finite morphisms, hyperplane sections all remain what they are after ¡ base extension. So, one has to show that π 1pZq is connected. As 58 Applications and Pathologies

¡ Z is ample on X (by deﬁnition), π 1pZq is ample on V (the pullback of an ample line bundle by a ﬁnite morphism between complete varieties is still ample). By Theorem 2.1, which we apply because ¡ dim V ¥ dim X ¥ 2 and because π 1pZq V is again ample, we can conclude.

Proof. (of Theorem 2.11) We may assume dim A d ¥ 2. As ãÑ n A is projective, we may fix a closed embedding A Pk . Let l denote the algebraic closure of k. Bertini says that there exists a n¦ P dense open subset U1 Pl such that every hyperplane H U1 gives a smooth connected (again by Therorem 2.1) intersection X p q H Al. Because k is infinite, U1 k is nonempty (it is even infinite). So one can choose a good hyperplane H1 with coordinates in k, and this will give a nonsingular (geometrically connected) variety X n ¡ n A H1 Pk , of dimension d 1. To this new subvariety of Pk (and to its induced embedding) one applies the same procedure, in n¦ order to find a new open subset U2 Pl and a new hyperplane X X n H2 defined over k, such that A H1 H2 Pk is nonsingular, of dimension d ¡ 2. In total, if one does so d ¡ 1 times, one ends with a nonsingular curve C A obtained by intersecting A with X ¤ ¤ ¤ X n a linear subspace ` H1 Hd¡1 Pk . Now, remembering Lemma 2.2, one discovers that for each nonsingular variety V with ¡ a finite morphism π : V Ñ A, one can say for sure that π 1pCq is geometrically connected. One has to use the above Lemma with smaller and smaller hyperplane sections: at the first step X ¡1p q Ñ one sets Z1 A H1, then one looks at π Z1 Z1 and sets X Z2 Z1 H2, and so on, until Zd¡1 C. Now, by the universal property of the map F : C ¢ C Ñ J, there is a morphism ψ : J Ñ A where J is the jacobian of C (note that C might not have a rational point). Let us call A1 the abelian subvariety ψpJq A. Let us assume, by contradiction, that A1 A. It is a theorem that A1 has a complement, that is, there exists an abelian subvariety A2 A such that A1 A2 A, the X ¢ Ñ intersection A1 A2 is finite and the map π1 : A1 A2 A sending p q ÞÑ P1, P2 P1 P2 is an isogeny (a surjective morphism with finite ¡1p q kernel), so in particular it is finite. The inverse image π1 A1 t p q P ¢ | P u P P, Q A1 A2 P Q A1 decomposes (note that Q A1) 7 p X q as a disjoint union of the following m : A1 A2 irreducible components: º ¡1p q p ¡ q ¢ t u π1 A1 A1 Q Q . P X Q A1 A2 ¡ ¡1p q Thus, if m 1, then because C A1 one has that π1 C is also disconnected, and this is a contradiction. But if m 1 (that is, π1 2.7. Arithmetic Bertini Theorem 59 is an isomorphism) one cannot conclude yet. However, one can choose an integer s coprime to char k, and consider the composition ¢ 1 s .. .. ¢ ...... ¢ p q...... p q A1 A2 . A1 A2 P, Q . P, sQ ...... π ...... 1 ...... π ...... A P sQ ¡1p q t p q P ¢ | P u In this case π A1 P, Q A1 A2 P sQ A1 says that P each such Q satisfies sQ A1. Now, because s is coprime to char k, the degree of 1 ¢ s is the power s2 dim A2 , and this is strictly bigger then 1. So, by the decomposition º ¡1p q p ¡ q ¢ t u π A1 A1 sQ Q P | P Q A2 sQ A1 one sees that there are deg p1 ¢ sq ¡ 1 components. Hence, as ¡1p q above, because C A1 one can conclude that neither π C is connected, contradiction.

Remark 2.5. The main result of this section remains true over finite fields, and the strategy of proof is exactly the same, provided that one applies Bertini’s Theorem for finite fields, replacing hyperplanes with hypersurfaces of higher degree (this is done in [18], and explained roughly in the Introduction.)

2.7 Arithmetic Bertini Theorem

The main object of study in this section will be arithmetic varieties over the ring of integers of a number field. One will start with some notations and definitions, in order to make some sim- ple but useful remarks. Definition 2.7.1. An arithmetic variety of dimension m is an integral scheme X, projective and flat over Z, such that the generic ¡ fiber XQ is regular of dimension m 1. If K is a number field and B Spec OK, an arithmetic variety over OK is a B-scheme X that is an arithmetic variety and whose generic fiber XK is geometrically irreducible over K. Proposition 2.7.1. Let Y be a Dedekind scheme and f : X Ñ Y a morphism, with X a reduced scheme. Then f is flat if and only if every irreducible component of X dominates Y (i.e. its image is dense in Y). 60 Applications and Pathologies

Proof. See [17], Proposition 3.9, p. 137.

Remark 2.6. Let us prove that an arithmetic variety X over OK is both projective and ﬂat over B Spec OK. Because X is projective n over Z, there is a closed immersion φ : X Ñ P for some n. The ¦ Z module M H0pX, φ Op1qq is ﬁnitely generated over Z (Theorem n 1 p q 1.5) so° one has a surjection Z M sending a0,..., an to the φ¦ps qa s P H0p n p qq sum j j j where j PZ , O 1 are the global sections determining φ. Thus one has a commutative triangle

θ ... X ...... PpMq ...... φ ...... n PZ and since φ is a closed immersion, so must be θ. Note that M is locally free over OK because it is projective. For ﬂatness one proceeds as follows. If f : X Ñ B is the structural morphism, there is a commutative triangle

f ...... XB...... σ ...... h ...... Spec Z where σ is separated and h is proper (because it is projective), so f is proper. Hence fpXq is either the whole B or it is one point. But if it is one point, then the same is true for hpXq, by commutativity of the triangle, and this is impossible as h is flat. So now apply Proposition 2.7.1 to the morphism f, to see that it must be flat. Indeed, fpXq B says that the unique irreducible component of X dominates B. Remark, finally, that an arithmetic OK-variety is necessarily horizontal, that is, it is flat and surjects onto the base.

The Arithmetic Bertini Theorem proved by Autissier in [2] is too advanced for our comprehension (it requires the theory of heights). So our final goal is to show the ”algebraic part”, which is however an important tool in the proof of Autissier main result. In the classical Bertini Theorem one calls good a hyperplane H giving a smooth intersection X X H. Recall that such a hyperplane section can be viewed as an element of the universal family n¦ ¢ p q ΣX Pk X, namely, as the set of those couples H, p where p 2.7. Arithmetic Bertini Theorem 61 ranges X X H. Next, in this arithmetic version, one will call good those hyperplanes with two properties instead of just one (we describe them below): this will make our good locus relatively small. ¡ Main characters. We fix a number field K and denote B Spec OK. This will be the base scheme, replacing the base field of classical Bertini Theorem. We also fix a locally free OK- module M of rank n 1 (this replaces the vector space kn 1). By P we mean the projective bundle PpMq Proj pSym Mq and by ¦ P its dual, both projective spaces of dimension n, not over a field but over OK (and, as usual, points in P have to be thought of as ¦ hyperplanes in P ). Let β : P Ñ B be the structural morphism and ¦ H P ¢B P the universal hyperplane. Our best friend will be an ¦ arithmetic OK-variety X P of dimension d ¥ 3, with structural ¡1 morphism f : X Ñ B. By Xb we will always mean the fiber f pbq. Note that these fibers are all of dimension d ¡ 1 (a flat morphism of irreducible algebraic varieties has fibers of constant dimension. See [Liu, Remark 3.15, p. 139] for the details); moreover, Xb is nonsingular over kpbq for all but finitely many b P B: indeed these bad points form a closed subset (of a scheme of dimension one), essentially by the openness condition in Theorem 1.9, and because saying that f is smooth at x P X amounts to asserting that f is flat at x and x is a smooth point in the fiber Xfpxq, over the field kpfpxqq. Finally, the bad b P B are collected in a finite locus J, that is:

J t b P B | Xb is singular over kpbq u B.

Here is a short preview of the future discussion: as usual, we are interested in answering questions like:

¥ is there a smooth hyperplane section X X H?

¥ how many hyperplane sections are smooth?

¡ Preview. For the first question: it turns out (Proposition 2.7.2) that what one can do without extending the base B is to find a closed horizontal subset Z P containing the bad locus (where we still have to specify what will be good/bad for us); in other words, we will be able to lock the bad hyperplanes inside a quite small ( closed) subset Z. This is already very nice, but it would count for nothing if there were no B-valued points inside the good locus (in that case, no good hyperplane would be defined over B): this of course can happen, because we do not work over a field, 62 Applications and Pathologies so an open subset of a projective space might have no rational points. Instead, what one can do by (suitably) extending the base 1 is to find a (arithmetic) hyperplane section X XOL over some 1 1 1 1 Ñ 1 B Spec OL such that the only singular fibers Xb1 of f : X B 1 correspond to those b above points in J (thus there are finitely many of them). For the second question: the answer is that one good hyperplane section is guaranteed if one fixes a suitable extension. If one wants a (infinite) family of good hyperplane sections one has to perform subsequent extensions K L1 L2 ...

Let us start with the details. First, let us ”visualize” the hy- ¦ perplanes inside P . These are exactly the ﬁbers Hy H ¢P ¦ Spec kpyq P of H Ñ P at all points y P P.

Another important object is the universal hyperplane section 1 ¢ ¦ X p ¢ q H H P X H P B X H (a universal family over P whose members are the sections X X Hy). If one calls π the 1 composition H ãÑ H Ñ P then π is the morphism deﬁning the 1 P 1 ¡1p q family H . More precisely, if y P is any point, then Hy π y coincides with the hyperplane section X X Hy.

In the sequel, one will work inside the open subset

t P | 1 u U0 y P π is ﬂat on Hy P.

¡ Interlude. Before going on, it is perhaps useful to set up a comparison with the usual Bertini Theorem, just to check the analogies.

Bertini over k Bertini over OK

n ¦ Main character: X Pk projective X P arithmetic

n¦ n ¦ Hyperplanes: Pk (H Pk ) P (Hy P )

X p n¦ ¢ q X p ¢ q 1 Univ. Hyp. Section: H Pk k X H P B X H

X n X 1 ¦ Hyperplane Sections: X H Pk X Hy Hy P

As it was announced, there is more than a single condition to deﬁne a good y P P. Let us see this in detail, by looking at the 2.7. Arithmetic Bertini Theorem 63 following arrows:

β . β ...... U0 ... B y . ... b ...... π . . f ...... 1 1 H X Hy Xb

One calls good those y P U0 such that: either p q P ¡ 1 one has β y B J, and Hy is smooth and geometrically irreducible over kpyq, or

P ¡1p q P p q p 1 q when y β b for b J, then in this case n Xb n Hy .

By resuming conditions and , one recovers a good locus ¤ 1 W W Y Wb U0 P bPJ

t P X ¡1p q | p q p 1 q u where Wb y U0 β b n Xb n Hy .

¡ Notation: If B is any scheme and f : X Ñ Y is a B-morphism, one will denote by fb the morphism f ¢ 1 : Xb Ñ Yb. A quasi-coherent sheaf F on X is said to be f-ﬂat at x P X in Ñ case Fx is ﬂat over OY,fpxq (via OY,fpxq OX,x). Let us recall the following result, whose proof can be found in [EGA IV3, p. 138].

Theorem 2.12 (Fibral Flatness Criterion). Let B, X, Y be locally noetherian schemes, F a quasi-coherent sheaf on X, f : X Ñ Y a B-morphism, and x P X. Assume that

...... h ...... Y ... B y . ... b ...... f ...... g ...... X x is commutative. Then, if F is coherent and Fx 0, the following are equivalent:

(1) The sheaf F is g-ﬂat at x and Fb is fb-ﬂat at x.

(2) The morphism h is ﬂat at y and F is f-ﬂat at x. 64 Applications and Pathologies

Coming back to our situation, there is a commutative square β ... P ...... B ...... π . . . . f . . . . 1 σ ... H ...... X

¡1 ¡ and if one denotes Pb β pbq t ηb u and τ f ¥ σ, one can use the criterion to show that for every b P B the generic point ηb ¡1 of Pb lies in U0: that is, π is ﬂat at every x P π pηbq. It is enough to consider

β ...... P ... B ηb . ... b ...... π ...... τ ...... H 1 x P P 1 p q for b B fixed and x Hb such that πb x ηb. So, taking the coherent sheaf F OH 1 , the statement says that the following are equivalent: 1 (1) The morphism Ñ 1 is flat, and π : H Ñ P is OB,b OHb,x b b b flat at x. Ñ (2) The morphism OB,b OPb,ηb is flat and π is flat at x. 1 Let us verify that (1) is true: H Ñ B is flat because it is a composition of flat morphisms (π is flat because f is and β is 1 flat because σ is; and σ is flat because H is a universal family). Ñ Finally, OPb,ηb Ox is flat as OPb,ηb is a field. Hence (2) is true, P 1 p q in particular π is flat at all points x Hb such that πb x ηb, which means exactly that ηb P U0. And this holds for all b P B. 1 Proposition 2.7.2 (Autissier). The locus P ¡ W is contained in a closed horizontal subset Z P, purely of codimension one. Hence 1 the good locus W contains the nonempty open subset P ¡ Z. 1 Proof. One has to show that ηb P W for all b P B. If this is 1 done, one can conclude because W is constructible (union of constructible subsets of P) and a constructible set which is a neighborhood of all the generic points contains an open subset. Start with b P B ¡ J. Let us note that f : X Ñ B is birational projective and B is normal so, according to Zariski Main Theorem (cfr. [Hartshorne, p. 280]) the fibers Xb of f are geometrically connected. Those coming from b P B ¡ J are also smooth over kpbq, 2.7. Arithmetic Bertini Theorem 65 so they are geometrically irreducible as kpbq-varieties. Hence, by H 1 classical Bertini’s Theorem, the generic hyperplane section ηb is smooth and geometrically irreducible over kpηbq (recall that dim Xb ¥ 2) and this says that ηb P W. b P J npH 1 q npX q η P W If then ηb b , so b b. Remark 2.7. Note that one can discard the second condition defin- 1 ing W , in the following sense: instead of considering the loci Wb one might take the whole fiber Pb for those y P U0 such that βpyq b P J. This is a situation more close to the usual Bertini Theorem but the good locus become quite larger, so it is morally 1 not so hard to find good hyperplanes. Instead, by considering W as above, one gets more information.

Theorem 2.13 (Autissier). There exists a ﬁnite extension L{K 1 1 and a closed subscheme X XOL such that, by letting B 1 Spec OL and g : B Ñ B, the morphism induced by the inclusion OK OL, one has: 1 • the scheme X is an arithmetic variety over OL, of dimension d ¡ 1; 1 P 1 p 1q P ¡ 1 • for every b B such that g b B J, the ﬁber Xb1 (of the 1 1 1 structural morphism f : X Ñ B ) is smooth and geometri- 1 cally irreducible over kpb q; 1 1 1 1 P p q P p q p 1 q • for every b B such that g b J, n Xb1 n Xgpb q .

Morally, Autissier’s theorem says that in the hyperplane sec- 1 tion X the only bad points come from ( live above) points of X which were already bad at the beginning. 66 Applications and Pathologies APPENDIX A

Appendix

This chapter is devoted to ﬁll the (main) gaps here and there left throughout the previous work. Giving complete proofs or definitions at the right moment would have diverted attention from the discussion.

Proof of Theorem 1.4

Theorem ¡ Let X be a variety and r a positive integer. Vector bundles of rank r over X correspond, up to isomorphism, to locally free OX-modules of rank r.

r r Proof. Let us use the identiﬁcation C A . Start with a complex vector bundle π : E Ñ X of rank r. Consider the sheaf Γp¡, Eq of sections of E, deﬁned as follows: for every open subset U X

s ΓpU, Eq t U −Ñ E | π ¥ s 1U u .

This amounts to saying that every point x P U is sent to a point of the C-vector space Ex. Note that ΓpU, Eq is an OXpUq-module: ﬁrst, it is an abelian group since we can add two sections s, t over U by declaring that ps tqpxq be the element spxq tpxq, sum of two vectors in Ex. We can also multiply by a regular function f : U Ñ 1 A , simply by evaluating: pfsqpxq fpxqspxq P Ex. It is easy to check the compatibility of restrictions. Now we show that Γp¡, Eq is locally free. Recall that, together with pE, πq, we are given an open covering pUiq of X on which E is trivial, i.e. we have a family | Ñ ¢ r r r s of trivializations E Ui − Ui A . If A Spec C x1,..., xr , we

67 68 Appendix can consider the local coordinate sections

r xi : Ui −Ñ Ui ¢ A

p −Þ Ñ pp, eiq where ei is the vector with all zeros except for a 1 in the i-th component. This is a section: when composed with the projection, we get the identity on Ui. Moreover, any section s P ΓpUi, Eq can ¤ ¤ ¤ be written as f1x1 frxr for some regular functions fi, so we can deﬁne a morphism of OXpUiq-modules

r ΓpUi, Eq −Ñ OXpUiq Þ Ñ p q s − f1,..., fr .

This is the required isomorphism. Note that we have been able to check the locally free property on the open covering on which E becomes trivial, just because these special subsets Ui speak to us and say: ”We trivialize E, so if you look at sections of the r projection Ui ¢ A Ñ Ui then you are done”. The way they say this is through the commutative diagrams ... r E| ...... ¢ Ui... Ui A ...... pr ...... π ...... Ui So, our argument simply says: such a section of pr is identiﬁed by a n-tuple of regular functions. Now start with a locally free OX-module F of rank r, and an open p q | | covering U Ui of X, so that F Ui is a free OX Ui -module of r i τ | r rank for every . Denote i the isomorphism F Ui OUi , and notice that U can be chosen to be ﬁnite, since² a variety is quasi- r compact. Construct the disjoint union F pUi ¢ A q, on which we will set an equivalence relation once we will have a suitable family of transition functions. For every i, j we have a couple of isomorphisms

τ τj i ... r ... r | ...... | ...... O F Ui OUi F Uj Uj and if we restrict each of them to Uij we get two new isomorphisms f i ... | ...... r ... O . F Uij ...... Uij fj 69

g f f¡1 P | r Hence, ij i j Aut F Uij Aut OUij . That is, for every i, j we can identify each gij to an r ¢ r matrix Aij with entries in OXpUijq. Call again gij the morphism of varieties Uij Ñ GLpr, Cq sending a point x P Uij to the matrix Aijpxq. These gij are tran- r sition functions, so we can glue the Ui ¢ C together along their r r intersections, by identifying px, vq P Ui¢C to any py, wq P Uj¢C whenever x y and w gijpxqv. The quotient of F that we get, together with the ﬁbration rpx, vqs ÞÑ x, is the required vector bundle on X.

Cˇ ech Cohomology

Let X be a topological space and U pUαqαPA an open covering. Put a well-ordering ¤ on A and define Σi, for i ¥ 0, to be r s the set of simplexes U Uα0 ,..., Uαi i.e. the set of those sets t u H Uα0 ,..., Uαi such that Uα0...αi . One can also interpret a simplex as a set of indices t α0,..., αi u such that α0 ¤ ¤ ¤ αi. For any (pre)sheaf F on X, define the (additive, say) group of Cechˇ i-cochains to be ¹ ¹ ip q p q p q C U, F F Uα0...αi F Uα0...αi ¤¤¤ α0 αi UPΣi ² In fact i-cochains are functions s : Σ Ñ ¤¤¤ pU q. i α0 αi F α0...αi For any i ¥ 0 there is a group homomorphism Bi : CipU, F q Ñ i 1p q p p qq ¤¤¤ C U, F defined by sending s s α0,..., αi α0 αi to the Bi P i 1p q p q t u function s C U, F taking the i 1 -simplex α0,..., αi 1 to the element

i¸ 1 k p¡1q spα , . . . ,α ˆ k,..., αi q|U 0 1 α0...αi 1 k0

One can show that Bi 1 ¥ Bi 0 so that C pU, F q is a complex. Let us make± some examples in low dimension: if i 0 then 0p q p q p p qq C U, F ²α F Uα , so that a 0-cochain s s α is a func- Ñ p q B0 B0 tion s : A α F Uα . Applying gives a 1-cochain s send- t u p q| ¡ p q| P ing the one-simplex α, β to the element s α Uαβ s β Uαβ F pUαβq. One veriﬁes that the transition functions of a vector bundle ˇ p ¢q are a Cech one-cochain² in the complex C U, OX . First, c p q Ñ ¢p q 1p ¢q ±gαβ : Σ1 α β OX Uαβ is an element of C U, OX ¢p q B1 α β OX Uαβ . It goes under to a function taking the two- simplex t α, β, γ u to the element p q p q¡1 p q| ¡1 | c β, γ c α, γ c α, β Uαβγ gβγgαγgαβ Uαβγ 70 Appendix

But the cocycle condition

gαγpxq gαβpxqgβγpxq @ x P Uαβγ says exactly that the c is a Cechˇ one-cocycle.

¦ ¦ ¦

Some relative constructions

Definition A.0.2. A morphism of schemes f : X Ñ Y is said to be ¡1 affine if there exists an open affine cover pYiq of Y such that f pYiq is affine for every i. If f is finite type then it is affine (in fact, finite type is equivalent to proper and affine). An affine morphism is quasi-compact and separated.

§ I. The Global Spec.

We start with the following Initial data: A noetherian scheme X and a quasi- coherent sheaf of OX-algebras A . That is: if U X is open, then A pUq is an OXpUq-algebra, and if U is afﬁne then A |U A pUq . In this situation, one proves that there exists a unique scheme over X, denoted Spec A , together with its structural morphism

β : Spec A −Ñ X , satisfying the following properties: for every open afﬁne subset ¡ U X, one has β 1pUq Spec A pUq, and for every inclusion ¡ ¡ V U of open afﬁne subsets of X, the inclusion β 1pVq ãÑ β 1pUq corresponds to the restriction A pUq Ñ A pVq. ¡ Translation: the object Spec A P SchpXq is constructed ad hoc to satisfy this: if U Spec B X is an open subscheme, then A pSpec Bq is some B-algebra AU, by our assumption on A . Well, Spec A is constructed so that above Spec B there lies exactly Spec AU...... Spec AU ...... Spec A ...... β . β ...... U Spec B ...... X 71

Note the analogy with vector bundles: the philosophy is again to construct a space by glueing ”prescribed” ﬁbers. The following result is a characterization of afﬁne morphisms.

Proposition A.0.3. If A is a quasi-coherent OX-algebra and β : Spec A Ñ X is the structural morphism, then β is afﬁne and Ñ A β¦OSpec A . Conversely, if f : Y X is afﬁne, then A : f¦OY is a quasi-coherent OX-algebra, and Y Spec A as X-schemes.

By the above Proposition, pSpec A , βq satisﬁes the following universal property:

UP: Given a morphism π : Y Ñ X, the X-morphisms Y Ñ Spec A are in Y-functorial bijection with morphisms απ making the following diagram commute:

... OX ...... π¦OY ...... απ ...... A

¡ Translation: If one considers the functor F : SchpXq −Ñ ShpXq sending the X-scheme π : Y Ñ X to π¦OY, then one is just asserting that the object β : Spec A Ñ X represents F, that is, F homXp¡, Spec A q. This natural isomorphism comes from the universal property that pSpec A , βq does satisfy. r Remark A.1. If X Spec B is afﬁne and A A for some B- algebra A, then the canonical morphism Spec A Ñ X satisﬁes the universal property above.

Proposition A.0.4. Start with a couple pX, A q with the usual assumptions. Assume there exists an object pSpec A , βq satisfying UP. Then, for every open subset U X, the morphism

β| U ... Spec A ¢X U pSpec A q|U ...... U satisﬁes the UP with respect to pU, A |Uq.

The latter Proposition, together with Remark A.1, shows that pSpec A , βq exists for all open subschemes U of any afﬁne scheme X Spec B. In fact, such an object exists in general. 72 Appendix

Remark A.2. In terms of the UP described above, one sees that the natural isomorphism A β¦OSpec A (let us call it φ) of Propo- sition A.0.3 comes from β ... X ...... Spec A ...... β ...... id ...... Spec A

That is, the identity map on Spec A over X corresponds to p¡ q β¦OSpec A under homX , Spec A . It follows that in a situation like in the diagrams

... π ...... YX OX ...... π¦OY ...... θ...... α ...... β ...... θ ...... φ ... id ... Spec A ...... Spec A A ...... β¦OSpec A one recovers α απ as the composition

φÑ Ñ A − β¦OSpec A − β¦θ¦OY π¦OY.

Good behavior under base change. Let f : Z Ñ X be any morphism, and A a quasi-coherent OX-algebra. There is a natural Z-isomorphism

¦ Z ¢X Spec A Spec f A .

This means that the canonical projection Z ¢X Spec A Ñ Z is ¦ exactly the structural morphism Spec f A Ñ Z.

Example A.0.1 (Total spaces and vector bundles). Let F be a locally free sheaf on X, of ﬁnite rank r. To F we can associate the (graded) sheaf of OX-algebras

¦ SpF q Sym F which (is also locally free and) assigns to each open subset U ¦ X the OXpUq-algebra Sym F pUq . Then Spec SpF q is a vector bundle on X: for every p P X there exists an open neighborhood ¡1p q p p qq| r U of p such that β U Spec S F U AU. This vector bundle is called the geometric vector bundle associated to F , and we denote it VpF q. Every vector bundle on X arises in this 73 way (see Theorem 1.4). Note that for every open subset U of X one has

r ¦ ΓpU p qq Γp r q pUqrx x s pUq , OSpec S F AU, OAU OX 1,..., r Sym F .

If, more generally, F is any quasi-coherent OX-module, then so ¦ is Sym F and thus the latter may be computed affine-locally ¦ (Proposition 1.1.2). In this case we call Spec Sym F the total space of F . n Example A.0.2. In the above example, take F OX (direct sum of n copies of OX). We define the trivial geometric vector bundle to be the X-scheme p nq n p nq¦ V OX AX Spec Sym OX . n r s 1 It generalizes the affine n-space over a ring AA Spec A x1,..., xn . Suppose we have a morphism f : X Ñ Y for some scheme Y. We p nq p nq¦ can consider the OY-algebra S OY Sym OY . Then, by what we said about the good behavior under base change, we have a n ¢ n natural isomorphism AX X Y AY : ¢ n ¢ p nq ¦ p nq X Y AY X Y Spec S OY Spec f S OY ¦p p nq¦q p nq¦ n Spec f Sym OY Spec Sym OX AX. ¦ ¦ We used that f commutes with Sym and that f OY OX.

§ II. The Global Proj.

Exactly in the same way as Spec specializes to Spec when X is afﬁne (see Remark A.1), we now perform a construction that will specialize to Proj of a graded ring when X is afﬁne. More precisely, we will generalizeÀ the following fact: when we take the Proj of a graded ring S d¥0 Sd, we get naturally a morphism Y Proj S Ñ Spec S0. We now replace Spec S0 by an arbitrary noetherian scheme X. To globalise our Proj we need also to replace the S0-algeba S by a sheaf of graded algebras S on X. At the end, we would like a structural morphism Proj S Ñ X.

Initial data: A noetherian scheme X andÀ a quasi- coherent sheaf of graded OX-algebras S d¥0 Sd, satisfying S0 OX. Then, for any afﬁneÀ open subset p q ` p q U Spec A X, we have S U A d¥1 Sd U .

1 n n n n¦ When X Spec A, of course AX AA, indeed AX Spec Sym OX r n p nq r s Spec Sym A Spec Sym A Spec A x1,..., xn , the last equality holding by Remark A.1. 74 Appendix

To construct Proj S Ñ X, let us ﬁx an afﬁne open subset U Spec A of X. Let us look at the scheme Proj S pUq and its structural morphism σU : Proj S pUq Ñ U. Now we use the quasi-coherence of S : for every distinguished inclusion Uf U, where Uf Spec Af for some f P A, we have an isomorphism

S pUq bA Af S pUqf S pUfq.

Hence, remembering that Proj commutes with tensor product,

p q p p q b q p q ¢ ¡1p q Proj S Uf Proj S U A Af Proj S U U Uf σU Uf .

In addition, for every inclusion U V of open subsets of X, the restriction S pVq Ñ S pUq coincides, in zero degree, with the restriction OXpVq Ñ OXpUq; this induces a commutative diagram

... Proj S pUq ...... Proj S pVq ...... σU . σV ...... UV...... Finally, one can check that for every two open afﬁne subsets U ¡1p X q ¡1p X q and V there is a natural isomorphism σU U V σV U V . All these information allow to glue the schemes Proj S pUq Ñ U together to get a global object

σ : Proj S Ñ X.

One can show that, by the construction, Proj S comes equipped with an invertible sheaf Op1q, which comes from glueing the var- ious Op1q on Proj S pUq.

Remark A.3. One often requires that S1 be coherent and generate S as an S0-algebra. In that case, any homogeneous part Sd is also coherent. To see this, fix an open affine subset U X. Then, for any d ¥ 1, the typical element of SdpUq is a product of d p q | elements of S1 U , so we see that Sd U is finitely generated over S0 OX. It follows (by Remark 1.1, for example, and using that X is noetherian) that Sd is coherent. One can show that if S1 is coherent, the structural morphism Proj S Ñ X is proper.

Proposition A.0.5. Let g : Z Ñ X be a morphism of noetherian schemes and let S be a quasi-coherent sheaf of graded OX- ¦ algebras as in our assumptions. Then Proj S ¢X Z Proj g S as Z-schemes. 75

We have a characterization of projective morphisms similar to the one we gave for affine morphisms. It is just the following definition. Definition A.0.3. A morphism of schemes f : X Ñ Y is said to be projective (and we say that X is projective over Y) if there is a Y-isomorphism X Proj S for some quasi-coherent sheaf of graded OY-algebras, which is generated in degree 1 over S0. Example A.0.3. For any scheme Y, the projective n-space over Y, n n ¢ Y defined to be PY PZ Spec Z , can be obtained as the global Proj p n 1q p n 1q of the sheaf S OY Sym OY : n p n 1q PY Proj S OY . Example A.0.4 (Projectivization and Projective Bundles2). Let X be a noetherian scheme and E a coherent OX-module. Define ¦ PpE q Proj SpE q Proj Sym E .

It is an X-scheme called the projectivization of E .

Example A.0.5. Any n-dimensional vector space V over a ﬁeld k may be viewed as a vector bundle V Ñ Spec k X. Writing V and ¦ V for the corresponding constant sheaves, we have their projec- tivizations, which are related but completely different spaces (as ¦ V and V themselves, in fact). The ﬁrst one, PpVq, is of course a projective space of dimension n, and its k-rational points correspond to one-dimensional quotients (hyperplanes) of V; dually, ¦ the k-rational points of PpV q correspond to one-dimensional subspaces of V. Summary: ¦ ¦ ¦ Pn PpVq Proj SpV q and Pn PpV q Proj SpVq.

In general, if E is a locally free OX-module of rank n 1, then PpE q is a projective bundle: above any afﬁne open subset U Spec A X there lies (under the structural morphism) exactly the n n U-scheme PU PA. n Remark A.4. Any closed subscheme Z PX can be written as Z Proj S , for some quasi-coherent sheaf S of graded OX- algebras. More generally, if E is coherent, any closed subscheme Z PpE q is the Proj of something; conversely, any quasi-coherent sheaf S of graded OX-algebras, generated by a coherent S1, is an epimorphic image of Sym S1, and such a surjection Sym S1 ãÑ p q S induces a closed immersion X Proj S P S1 .

2 See Example A.0.1. 76 Appendix

Finally, a comparison between Spec and Proj: a projective bundle is a particular case of the projectivization construction, exactly like a vector bundle is a particular case of a total space (the only observation here is ’locally free of ﬁnite rank’ ñ ’coherent’).

The Frobenius morphism

Let k be any ﬁeld of characteristic p ¡ 0 and denote by φ : k Ñ k the Frobenius endomorphism sending x ÞÑ xp. For an algebraic variety X over k, consider the map f : X Ñ X which is the identity on points and, at the level of structure sheaves, is deﬁned as follows:

7 .... f pUq : OXpUq ...... OXpUq

...... p a . ... a . 7 Note that FX pf, f q is a morphism of ringed spaces but not 7 a morphism of k-varieties, unless k Fp, because f might not preserve the structure of k-algebra. FX is called the absolute Frobenius of X; note that it commutes with any morphism f : X Ñ Y of k-schemes, in the sense that f¥FX FY ¥f. We now change the structure of k-algebra on k to make FX into a k-morphism. To do 1 so, call k the field k whose structure of k-algebra is defined by a 1 new multiplication by scalars, namely, for every λ P k and x P k , we set λ ¤ x φpλqx λpx. For a finitely generated k-algebra A, define a k-algebra structure on

ppq 1 A A bk k by letting λpa b µq a b pλµq, and note that the map

ppq ...... ρ : A .... A ...... p a b µ . ... µa is a k-morphism, indeed, ρpλpa b µqq ρpa b pλµqq λρpa b µq. ppq ppq ppq For X Spm A and X Spm A , deﬁne FX{k : X Ñ X to be the k-morphism corresponding to ρ. It is called the relative r s ppq Frobenius of X. For example, if A k°x1,..., xn ,° then ρ : A Ñ b ÞÑ p ν ÞÑ pν A A sends xi 1 xi and hence ν aνx ν aνx .

We now define the relative Frobenius of an arbitrary algebraic variety X. In what follows S denotes the base variety Spm k 1 ppq Spm k and σ : X Ñ S the structural morphism. Define X Xk1 . 1 Defining the new structure of k-algebra on k is the same (in the 77

1 category of schemes over S) as viewing k Ñ k as a base change, relative to which we have a pullback diagram

ppq ...... π ...... X X ¢S S ... X ...... σ ...... SS...... Moreover, coming back to the absolute Frobenius of X, we know that the following square commutes: F X ... XX...... σ . . σ ...... F ... S ... SS...... Putting all this together, we use the universal property of pullback in the diagram

...... X ...... σ ...... FX ...... FX{k ... ppq ...... X S X ...... σ ...... FS ...... S to deduce the existence of a unique canonical morphism FX{k : p q X Ñ X p , making all commutative. We call it the relative Frobe- nius of X. If pXiq is an open afﬁne covering of X, then FX{k results Ñ from the glueing of the afﬁne relative morphisms FXi{k : Xi ppq ÞÑ ppq Xi . Note that, because pullback is functorial, so is X X . In fact, for any k-morphism f : X Ñ Y there is a commutative diagram

f ... XY...... F { . . F { X k . . Y k ...... ppq ... p q f .... p q X p ...... Y p p q Fix a ﬁeld extension K{k. Then the functor ¡ p commutes with ppq ppq this base extension, i.e. pX qK pXKq . Moreover FX{k induces p q Ñ ppqp q n n a map on rational points X K X K . If X is Ak or Pk then 78 Appendix

p q ÞÑ p p p q p q ÞÑ this map is simply t1,..., tn t1 ,..., tn , or t0,..., tn p p p q t0 ,..., tn , defined on K-rational points. In particular, by reduc- n ing to the standard affine covering° on X° Pk , we see that FX{k is r s ν ÞÑ pν defined on k x0,..., xn by ν aνx ν aνx . p q k Lemma A.1. Let X V I An be an algebraic variety over ppq r s{ ppq ppq k Fp. Then X Spm k °x1,..., xn I where° I is the ideal p ν ν P generated by polynomials ν aνx , where ν aνx I. The Ñ ppq ÞÑ p relative Frobenius FX{k : X X is induced by xi xi . Proof. See [17], Lemma 2.25, p. 95.

Corollary 3. Let X be an algebraic variety over k Fp. ppq (a) We have X X and FX FX{k (absolute and relative Frobenius coincide).

(b) Let k be the algebraic closure of k. Let FX : Xpkq Ñ Xpkq be the map induced by composition with FX. If X VpIq is a n r s closed subvariety of Ak Spm k x1,..., xn , then we have p q p p p q FX t1,..., tn t1 ,..., tn . Proof. See [17], Corollary 2.26, p. 95. Bibliography

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